六角形自举法揭示N=4超杨-米尔斯理论中HHL三点函数的L依赖性

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本文主要探讨的是对角线形状因子和六角形形状因子在平面N=4超杨-米尔斯理论中的应用，特别是在计算重轻（HHL）三点函数时。最近，Bajnok、Janik和Wereszczynski提出了一个用于研究量子场论的六边形自举程序，这是一种创新的方法，旨在通过解析手段处理AdS/CFT对偶中的积分方程，从而得到理论中的物理量如结构常数。作者Yunfeng Ji和Andrei Petrovski使用这个新颖的六边形bootstrap程序，分析了HHL三点函数的特性，这些函数在理论中代表了不同能量级别的相互作用。他们的工作重点在于验证关于这些函数的一个猜想，即在重算子长度L下，其结构常数具有多项式L依赖性，并且这个依赖关系一直持续到主导的有限尺寸校正。L值在这里表示重态操作员的尺度，校正是对理论预测与实际观测之间的微小偏差的修正。虽然论文的证明是基于特定的设置，但其方法的普适性被强调，意味着它能够扩展到更广泛的情况，这对于理解和解决更复杂的理论问题具有重要意义。论文涉及到的关键概念包括AdS/CFT对应关系、可积场论、超对称性和对偶性，这些都是现代理论物理学中研究弦理论和量子引力的重要工具。值得注意的是，该研究发表在《Journal of High Energy Physics》(JHEP)上，属于开放存取资源，由Springer出版，并于2016年7月发布。作者还提到了文章是在arXiv上预印本的版本，编号为1511.06199，并且得到了SCOAP3项目的资金支持。引用时应给出doi:10.1007/JHEP07(2016)120。这篇论文不仅贡献了对角线和六角形形状因子在N=4超杨-米尔斯理论中的新理解，而且还展示了如何通过创新方法处理复杂的对称性和非perturbative效应，对于深化我们对量子场论和强相互作用的理解具有重要的科学价值。

JHEP07(2016)120

2 The hexagon bootstrap program

We review brieﬂy the proposal by Basso, Komatsu and Vieira [1]. Intuitively, the structure

constant can be represented by a pair of pants either in the spin chain or string theory

description. The main idea of [1] is cutting this pair of pants into two objects called the

hexagons or the hexagon form factors. When cutting the pair of pants, the excitations on

each operator can be attributed to either hexagon and one needs to sum over all the pos-

sible partitions, each partition is associated with certain weight. In addition, gluing back

the two hexagons into a pair of pants requires summing over all possible states living on

the three gluing segments, which can be performed by means of integrating over the mirror

excitations. The calculation of the hexagons can be achieved in two steps. The ﬁrst step is

to move all excitations on the same edge by performing mirror transformations. The result-

ing object is called fundamental hexagon. It can be denoted as H

···A

, · · · , u

where A

are the su

(2|2) ⊗ su

(2|2) bifundamental indices for the i-th excitation and

is the corresponding rapidity. The second step is the computation of the fundamental

hexagon itself which is given by the following prescription

H = H

dyn

· H

mat

, (2.1)

where

dyn

i<j

h(u

, u

) (2.2)

mat

= (−1)

hχ

· · · χ

|S|χ

· · · χ

and the factor (−1)

accommodates for the grading. For our case below, we consider only

scalar excitations and f = 0. The dynamical part is simply a product of the scalar function

h(u, v) given by

−

− x

−

− x

1 − 1/x

−

1 − 1/x

(2.3)

where the variables x

1,2

are deﬁned as x

= x(u ± i/2) and x

= x(v ± i/2). Here x(u)

is the Zhukowsky variable satisfying x + 1/x = u/g and σ

= σ(u, v) is the square root of

BES dressing phase [8].

For the matrix part, χ

denotes a state in the fundamental representation of su(2|2)

and S is Beisert’s S-matrix [9, 10]. The matrix part of the hexagon form factor is thus

given by a factorized product of Beisert’s S-matrix elements with the dressing phase set

to 1.

3 Set-up

We study the HHL structure constant with the excitations of the two heavy operators being

the transverse scalar excitations X = Φ

and

X = −Φ

and the light operator being the

BPS operator, which is also called the reservoir in [1]:

= tr

Z = Z +

Z + Y −

Y . (3.1)

– 2 –

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