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
1
2
open spin-chain Hamiltonian
with Neumann boundary conditions [53]. Let us briefly review some important results
in [54] before our computations start.
19
For general boundary conditions, the open spin-chain Hamiltonian appearing in the
Wilson loops can be expressed as,
H =
L−1
X
k=1
(I
k,k+1
− P
k,k+1
) + C
1
(I − Q
φ
6
1
) + C
L
(I − Q
φ
6
L
), (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
φ
6
1
and Q
φ
6
L
appeared since we choose the scalar in the Wilson loop as φ
6
. As a result, Q
φ
6
1
and Q
φ
6
L
are defined as
20
Q
φ
6
1
|φ
6
· · · i = 0, Q
φ
6
1
|Z · · · i = |Z · · · i,
Q
φ
6
L
| · · · φ
6
i = 0, Q
φ
6
L
| · · · Zi = | · · · Zi. (3.2)
19
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.
20
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 –