基于积分性的Wilson环算子结构常数：强耦合验证与全息对应

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本文主要探讨的是N=4超杨-米尔斯理论中的一个重要概念——基于可积性威尔逊环上的算子结构常数。结构常数是量子场论中的关键量，它描述了在特定背景下，如 Wilson 环（一个封闭路径上包围着物理系统的线圈，通常用于探测强相互作用）上插入的局部算子之间的相互作用强度。在这个特定理论中，结构常数被猜想为一个六边形的几何表示，其中三个边界被边界状态压缩，这反映了理论的对称性和镜像对称性。作者们提出了一种有限耦合表达式，这个表达式与全息描述相吻合，后者通过一个六角形斑块和三个边界构成的立方开放字符串顶点来描述。全息原理是一种将弦理论中的物理现象映射到引力理论中的现象，这里的全息描述提供了一个更为直观和深层次的物理图像。文章的核心内容集中在弱耦合区域的验证上。在弱耦合情况下，结构常数的渐近行为简化为对六边形形状因子中所有可能改变磁矩符号的方式的和的计算。这涉及到对 Wilson 循环和开放自旋链上的算符之间关系的利用，这些关系在量子色动力学的树级计算中起着关键作用。作者们在 SU(2) 板块中计算出的结果与他们的猜想完美匹配，显示了理论在不同能量尺度下的稳健性。这篇论文不仅探讨了理论物理学中的一个核心问题，而且还展示了从量子场论到全息描述的桥梁，以及如何通过数学方法（如可积性）来解析复杂理论中的结构常数。这对于理解 N=4 超杨-米尔斯理论以及其与弦理论的联系具有重要的意义，也为未来的研究提供了新的视角和计算工具。

JHEP11(2017)116

with respect to rapidity variables u

(not the momentum), and by the propagation factor

for the length of the operator O

. Also, we deﬁne

(M)

∪P

−

={1,...,M}





k∈P

−



−e

2ip

)



l<k

S(p

, p

)S(−p

, p

)





i<j

Y Y

(ˆp

, ˆp

)

(2.3)

with

ˆp

(

i ∈ P

−p

i ∈ P

−

. (2.4)

Note that `

≡ (L

+ L

− L

)/2 (i, j and k are all diﬀerent ) is the bridge length between

the operator O

and O

. Thus, in the numerator there is a propagation factor for the

bridge length between the operator O

and O

. The B

(p) is the reﬂection amplitude at

ﬁnite coupling and h

Y Y

is the two-particle hexagon form factor at the ﬁnite coupling.

the case of two magnon, the above expression is simply written as

(2)

= h

Y Y

(u, v) − S(p

, p

)S(−p

, p

2ip

Y Y

(−u, v) (2.5)

− B

2ip

Y Y

(u, −v) + S(p

, p

)S(p

, −p

2i(p

Y Y

(−u, −v).

Even for the other conﬁgurations such as C

11◦

123

, the expression at the ﬁnite coupling

can be managed in similar way. The rule is that the structure constant is given by the

summation over the sign ﬂipping hexagon form factor with negative sign and appropriate

factors related dynamical processes of the magnons.

Since the hexagon form factor itself was determined at ﬁnite coupling by symmetry

and an integrable bootstrap method [31], the spin-chain analysis should be reproduced

by considering a weak coupling expansion of the hexagon form factor. In our setup, the

inserted operators are interpreted as three open spin-chains with excitations which are

realized as magnons on physical edges of the hexagon. Surely, to manage the boundary

information of the hexagon is a nontrivial part. However, we could actually check that

there is no new object for the bootstrap method at least up to SU(2) sectors and the

leading order of the weak coupling.

Furthermore, since the exact reﬂection matrix R(p)

is already available from [48, 49],

the generalization can be straightforwardly performed.

The multi-magnon hexagon form factor can be factorized into two-body hexagon form factor.

Honestly it does not guarantee that such a new object is always unnecessary for integrable bootstrap

of open string correlators. For example, the higher rank sector of the Wilson loops with operator insertions

or other integrable open strings such as maximal giant gravitons may have some nontrivial contributions

from the boundary. To clarify this point would deﬁnitely be an invaluable future direction.

At the weak coupling of SU(2) sector, R(p) just becomes a phase R(p) = e

. As we shall analyze

later, our reﬂection phase B

(p) is not the conventional reﬂection phase R(p) since we shall use the half-

step shifted basis [54]. In usual, the one-magnon wave function in an open-chain is written as ψ

(1)

(x) =

α(p)e

ipx

+ α

(p)e

−ipx

. The conventional left reﬂection phase is deﬁned by R

left

(p) ≡

(p)

α(p)

. Then, we have

(1)

(x) = α(p)



ipx

+ R

left

(p)e

−ipx



. On the other hand, our boundary S-matrix B

(p) is related to the

reﬂection matrix R

left

(p) as R

left

(p) = B

(p)e

2ipL

. Thus, the reﬂection phase corresponding to the SU(2)

part of the exact reﬂection matrix in [48, 49] becomes the Neumann boundary condition which is realized

with B

(p) = 1.

– 6 –

JHEP11(2017)116

In next section, we shall compute structure constants at tree level of the SU(2) sector.

Unlike at higher-loops or in full sectors, we need not to consider the nested procedure

through the nondiagonal reﬂection matrix with the nontrivial dressing phase and we can

understand their dynamical information well. If we now consider the weak coupling limit

of (2.1), the diﬀerence is just the coupling dependence. Namely the S-matrix, the Bethe-

Yang equation, the reﬂection matrix and the hexagon form factor should be replaced by

their weak coupling forms. We shall explicitly check this by computing tree-level structure

constants through open chain Bethe wavefunctions.

3 Tree-level analysis for the SU(2) sector

In previous section, we gave a conjecture for asymptotically exact structure constants of

local operators on Wilson loops. At the leading order of weak coupling regime, it reduces to

the tree-level three-point functions from open spin-chains. To check it, we shall explicitly

calculate the correlation functions of three operators corresponding to open spin-chains

at the tree-level of the SU(2) sector. Since the scalar excitations in this sector are not

overlapped with the scalar in the Wilson loops, one can just consider contractions between

the excitation ﬁelds in the inserted operators realized by open spin chains.

3.1 Coordinate Bethe ansatz for open spin-chain

According to [52], the one-loop anomalous dimension of the planar N = 4 super Yang-Mills

corresponds to an integrable spin-chain Hamiltonian, and the spin-chain state is given by an

eigenfunction obtained by diagonalizing the Hamiltonian. The one-loop anomalous dimen-

sion for our setup is given as the eigenvalue of the XXX

open spin-chain Hamiltonian

with Neumann boundary conditions [53]. Let us brieﬂy review some important results

in [54] before our computations start.

For general boundary conditions, the open spin-chain Hamiltonian appearing in the

Wilson loops can be expressed as,

H =

L−1

k=1

k,k+1

− P

k,k+1

) + C

(I − Q

) + C

(I − Q

), (3.1)

where I

k,k+1

is the identity operator in ﬂavor-space and P

k,k+1

is the permutation operator

which switches two scalars at site k and at site k + 1 each other. Here, the operators Q

and Q

appeared since we choose the scalar in the Wilson loop as φ

. As a result, Q

and Q

are deﬁned as

|φ

· · · i = 0, Q

|Z · · · i = |Z · · · i,

| · · · φ

i = 0, Q

| · · · Zi = | · · · Zi. (3.2)

We shall use slightly diﬀerent conventions to [54]. In our notation, processes of the magnons on

“coordinate axes” become more clear. Thus, when we compare the tree level results to our conjecture, it

would be better to use. The relation in detail is introduced in appendix A.

The notation was introduced in [46]. In addition, although we shall not introduce the computations,

we actually have calculated the SO(6) Hamiltonian by evaluating all Feynman diagrams, and could check

that the result is written as the above form.

– 7 –

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