In this article, we shall develop and formulate two novel viewpoints and properties
concerning the three-point functions at weak coupling in the SU(2) sector of the

Article funded by SCOAP3

\equiv \left(\prod_{k=1}^{\ell}\bra{\s_{k; \ell+1-k}}\right)\ket{\Psi_1} \otimes \ket{\Psi_2}\period\label{eq-17} } Here $\ell$ is the length of the spin chain and $\bra{\s_{k; \ell+1-k}}$ is the state which projects out the singlet part made out of the spin state at $k$-th site of $\ket{\Psi_1}$ and the one at the $(\ell+1-k)$-th site of $\ket{\Psi_2}$. The operation should be quite clear from figure~\ref{2pt_comp}. \begin{figure} \centering \includegraphics[clip,height=3cm]{2pt_comp-m.png} ]]>

\big<\ket{\tilde{\mathcal{O}}_1}_R\comma\ket{\tilde{\mathcal{O}}_2}_R \big>\period\label{eq-18} } As is manifest in~\eqref{eq-18}, the Wick contraction between two operators factorizes into the part coming from the \SU(2)$_L$ chain and the part coming from the \SU(2)$_R$ chain. This factorization property continues to hold for the tree-level three-point functions, since they are computed through the contractions between the composite operators in the manner described above. ]]>

\equiv \sum_{a,b,c}\big< \ket{\mathcal{O}_{1_a}}^{r}\comma \ket{\mathcal{O}_{2_b}}^{l}\big>\big< \ket{\mathcal{O}_{2_b}}^{r}\comma \ket{\mathcal{O}_{3_c}}^{l}\big>\big< \ket{\mathcal{O}_{3_c}}^{r}\period \ket{\mathcal{O}_{1_a}}^{l}\big>\period\label{eq-21} } Multiplying the contribution from the \SU(2)$_R$ sector, which is entirely similar to~\eqref{eq-21}, the final formal expression for the structure constant is given by \beq{ C_{123}=\frac{\sqrt{\ell_1 \ell_2 \ell_3}}{N_c\sqrt{\mathcal{N}_1\mathcal{N}_2\mathcal{N}_3}}\big< \ket{\mathcal{O}_1}_L\comma \ket{\mathcal{O}_2}_L \comma\ket{\mathcal{O}_3}_L \big>\big< \ket{\tilde{\mathcal{O}}_1}_R\comma \ket{\tilde{\mathcal{O}}_2}_R \comma\ket{\tilde{\mathcal{O}}_3}_R \big>\comma \label{eq-22} } where $\mathcal{N}_k$ denotes a factor coming from the normalization of the operator $\mathcal{O}_k$. \begin{figure} \centering \includegraphics[clip,height=3cm]{3pt_comp-m.png} ]]>

=\big< \ket{\psi_1}\comma \sigma_2\, \Omega_2^{t} (u)\,\sigma_2 \ket{\psi_2}\big>\period\label{eq-37} } Here, again $t$ and $\sigma_2$ act on the auxiliary space and $\Omega_{n}$ is the monodromy matrix acting on $\ket{\psi_n}$ defined by \beq{ \begin{aligned} &\Omega_1(u) = {\rm L}_1(u-\theta^{(1)}_1)\cdots {\rm L}_{\ell}(u-\theta^{(1)}_\ell)\comma\quad \Omega_2(u) = {\rm L}_1(u-\theta^{(2)}_1)\cdots {\rm L}_{\ell}(u-\theta^{(2)}_\ell)\period \end{aligned}\label{defombra} } The parameters $\theta^{(n)}$'s are the inhomogeneities for $\ket{\psi_n}$. In order for~\eqref{eq-37} to be satisfied, we need to make the following identification between the inhomogeneities (see figure~\ref{inhomo}): \beq{ \theta^{(1)}_k = \theta^{(2)}_{\ell-k+1}\period\label{inhomoeq} } In terms of the Wick contraction in the gauge theory, this amounts to assigning the same inhomogeneity parameter to each two spin sites contracted by a propagator. This is precisely the identification we need when we study the one-loop correction using the inhomogeneities~\cite{Tailoring4, Kostov1, Kostov2, Fixing} and we impose such relation throughout this paper. \begin{figure} \centering \begin{subfigure}[b]{0.38\textwidth} \centering \includegraphics[width=\textwidth]{inhomo1-m.png} ]]>

\comma
}
where, for definiteness, we wrote down the indices for the auxiliary space.
Since the $k$-th site of $\ket{\psi_1}$ is contracted with the $(\ell-k+1)$-th site of $\ket{\psi_2}$ as shown in figure~\ref{inhomo} and the inhomogeneities are identified as~\eqref{inhomoeq}, the Lax operator transforms as follows under the application of the crossing relation~\eqref{crossed1}:
\beq{
\left( {\rm L}_k (u-\theta^{(1)}_k)\right)_{i_ki_{k+1}} \to \quad \left( \sigma_2 {\rm L}^{t}_{\ell-k+1}(u-\theta^{(2)}_{\ell-k+1})\sigma_2\right)_{i_k i_{k+1}}\period
}
Then, moving the Lax operators one by one, we obtain
\beq{
\begin{aligned}
&\big< \ket{\psi_1}\comma \left( \sig_2 {\rm L}_{\ell}^t(u-\theta^{(2)}_{\ell}) \sig_2\right)_{i_1i_2}
\left( \sig_2 {\rm L}_{\ell-1}^t(u-\theta^{(2)}_{\ell-1}) \sig_2\right)_{i_2i_3}\cdots \left( \sig_2 {\rm L}_{1}^t(u-\theta^{(2)}_{1}) \sig_2\right)_{i_{\ell} i_{\ell+1}} \ket{\psi_2}\big> \\
& = \big< \ket{\psi_1} \comma \left( \sig_2 ( {\rm L}_{1}(u-\theta^{(2)}_{1})\cdots {\rm L}_{\ell}(u-\theta^{(2)}_{\ell}) )^t \sig_2\right)_{i_1 i_{\ell+1}} \ket{\psi_2} \big>\\
&= \big< \ket{\psi_1}\comma \left( \sigma_2 \Omega_2 ^t(u)\sigma_2\right)_{i_1i_{\ell+1}} \ket{\psi_2}\big>\period
\end{aligned}
\label{eq-38}
}
In terms of components, $\sigma_2 \Omega^t_2(u) \sigma_2$ is given by
\beq{
\sigma_2\Omega_{2}^t (u)\sigma_2=\sigma_2\pmatrix{cc}{A^{(2)}(u)&B^{(2)}(u)\\C^{(2)}(u)&D^{(2)}(u)}^t\sigma_2=\pmatrix{cc}{D^{(2)}(u)&-B^{(2)}(u)\\-C^{(2)}(u)&A^{(2)}(u)}\period
}
Here and throughout this subsection we put superscripts $(1)$ or $(2)$ in order to distinguish the components of $\Omega_1$ from those of $\Omega_2$.
The formula~\eqref{eq-37} in particular contains the crucial relation
\beq{
\big**=-\big<\ket{\psi_1}\comma B^{(2)}(u)\ket{\psi_2}\big>\comma\label{eq-39}
}
which only involves $B(u)$ operators. In the $u\to\infty$ limit, the relation~\eqref{eq-39} produces
\beq{
\big =-\big<\ket{\psi_1}\comma S^{(2)}_{-}\ket{\psi_2}\big>\period\label{s-relation}
}
Then, by a repeated use of~\eqref{eq-39} and~\eqref{s-relation}, we can collect all $B(u)$'s and $S_{-}$'s on one side and transform the skew-symmetric inner product which appear in the sewing procedure, such as
\beq{
\big< e^{xS^{(1)}_{-}}B^{(1)}(u_1)\cdots B^{(1)}(u_{M_1})\ket{\upspin^{\ell}}\comma e^{yS^{(2)}_{-}}B^{(2)}(v_1)\cdots B^{(2)}(v_{M_2})\ket{\upspin^{\ell}}\big> \comma
}
into the following expression:
\beq{
(-1)^{M_1}\big< \ket{\upspin^{\ell}}\comma e^{(y-x)S^{(2)}_{-}}B^{(2)}(u_1)\cdots B^{(2)}(u_{M_1})B^{(2)}(v_1)\cdots B^{(2)}(v_{M_2})\ket{\upspin^{\ell}}\big>\period\label{eq-40}
}
From the definition of the skew-symmetric inner product~\eqref{eq-17}, this expression can be readily evaluated\footnote{Essentially, due to the
skew-symmetry, each time the singlet projector acts on a pair of spins, the up-spin is converted to the down-spin and this produces $\bra{\downarrow^\ell\!\!}$.}
as a matrix element in the spin-chain Hilbert space as follows:
\beq{
\begin{aligned}
&(-1)^{M_1}\big< \ket{\upspin^{\ell}}\comma e^{(y-x)S^{(2)}_{-}} B^{(2)}(u_1)\cdots B^{(2)}(u_{M_1})B^{(2)}(v_1)\cdots B^{(2)}(v_{M_2})\ket{\upspin^{\ell}}\big>\\
& =(-1)^{M_1}\bra{\downarrow^{\ell}\!}e^{(y-x)S^{(2)}_{-}}B^{(2)}(u_1)\cdots B^{(2)}(u_{M_1})B^{(2)}(v_1)\cdots B^{(2)}(v_{M_2})\ket{\upspin^{\ell}}\period
\end{aligned}\label{eq-41}
}
It is important to recognize that a matrix element of the form~\eqref{eq-41} can be identified with the so-called partial domain wall partition function. More precisely, we can show
\beq{
\bra{\downarrow^{\ell}\!}e^{zS_{-}}B(x_1)\cdots B(x_{M})\ket{\upspin^{\ell}}=z^{\ell-M}Z_{p}(\bm{x}|\bm{\theta})\comma\label{funda}
}
where $Z_{p}(\bm{x}|\bm{\theta})$ is the partial domain wall partition function (pDWPF), which is given by~\cite{Kostov1,Kostov2}:
\begin{align}
\label{eq-43}
Z_{p}\left(\bm{x}|\bm{\theta}\right)& \equiv{\frac{1}{(\ell-M)!}}\bra{\downarrow^{\ell}\!}S_{-}^{\ell-M}B(x_1)\cdots B(x_{M})\ket{\upspin^{\ell}}\\
&=\frac{\prod_{i=1}^{M}\prod_{j=1}^{\ell} (x_i-\theta_j-i/2)}{\prod_{i**

$. First, using the coset parametrization~\eqref{eq-13}, each spin-chain state can be expressed~as \beq{ \begin{aligned} &\ket{\mathcal{O}_1}_L=\left( \frac{1}{1+|z_1|^2}\right)^{\ell_1/2 -M_1}e^{z_1 S_{-}}\ket{\bm{u}^{(1)};\upspin^{\ell_1}}\comma\\ &\ket{\mathcal{O}_2}_L=\left( \frac{1}{1+|z_2|^2}\right)^{\ell_2/2 -M_2}e^{z_2 S_{-}}\ket{\bm{u}^{(2)};\upspin^{\ell_2}}\comma\\ &\ket{\mathcal{O}_3}_L=\left( \frac{1}{1+|z_3|^2}\right)^{\ell_3/2 -M_3}e^{z_3 S_{-}}\ket{\bm{u}^{(3)};\upspin^{\ell_3}}\comma \end{aligned}\label{3ptstates} } where $\bm{u}^{(k)}$ denotes the set of rapidities for the operator $\mathcal{O}_k$ and its number of elements is denoted by $M_k$. Then, we can apply the formula~\eqref{eq-35} to split each chain into two and compute the skew-symmetric inner product using~\eqref{funda}. When computing the inner product, it is important that we assign the same inhomogeneity parameter to any two spin sites contracted by a propagator as discussed in the previous subsection. In the current setup, this leads to the following relation among the sets of inhomogeneities (see figure~\ref{inhomo}): \beq{ \bm{\theta}^{(1)}=\bm{\theta}^{(31)}\cup\bm{\theta}^{(12)}\comma \quad \bm{\theta}^{(2)}=\bm{\theta}^{(12)}\cup\bm{\theta}^{(23)}\comma \quad \bm{\theta}^{(3)}=\bm{\theta}^{(23)}\cup\bm{\theta}^{(31)}\comma\label{setinhomo} } where $\bm{\theta}^{(n)}$ is the set of inhomogeneities for $\ket{\mathcal{O}_n}_L$ and $\bm{\theta}^{(nm)}$ denote the set of the inhomogeneities common to $\ket{\mathcal{O}_n}_L$ and $\ket{\mathcal{O}_m}_L$. As a result of these operations, we obtain the following final form expressed in terms of the sum-over-partitions \begin{align} &\big< \ket{\mathcal{O}_1}_L\comma \ket{\mathcal{O}_2}_L\comma \ket{\mathcal{O}_3}_L\big>\notag\\ &=\left( \frac{1}{1+|z_1|^2}\right)^{\ell_1/2 -M_1}\left( \frac{1}{1+|z_2|^2}\right)^{\ell_2/2 -M_2}\left( \frac{1}{1+|z_3|^2}\right)^{\ell_3/2 -M_3}\label{generalformula}\\\notag &\qquad\times \sum_{\bm{\alpha}_l^{(k)}\cup\bm{\alpha}_r^{(k)}=\bm{u}^{(k)} } z_{21}^{\ell_{12}-|\bm{\alpha}_r^{(1)}|-|\bm{\alpha}_l^{(2)}|}z_{32}^{\ell_{23}-|\bm{\alpha}_r^{(2)}|-|\bm{\alpha}_l^{(3)}|}z_{13}^{\ell_{31}-|\bm{\alpha}_r^{(3)}|-|\bm{\alpha}_l^{(1)}|} \, \mathcal{D}_{\{\bm{\alpha}_{l,r}^{(1)},\bm{\alpha}_{l,r}^{(2)}\bm{\alpha}_{l,r}^{(3)}\}}\comma \end{align} In this expression, $|\bm{\alpha}_{l,r}^{(k)}|$ stands for the number of elements of $\bm{\alpha}_{l,r}^{(k)}$ and $z_{nm}$ denotes the difference $z_n-z_m$. The last factor $\mathcal{D}_{\{\bm{\alpha}_{l,r}^{(1)},\bm{\alpha}_{l,r}^{(2)}\bm{\alpha}_{l,r}^{(3)}\}}$, which is independent of the polarizations, is given in terms of the pDWPF as \begin{align} \mathcal{D}_{\{\bm{\alpha}_{l,r}^{(1)},\bm{\alpha}_{l,r}^{(2)}\bm{\alpha}_{l,r}^{(3)}\}}&\equiv (-1)^{|\bm{\alpha}_r^{(1)}|+|\bm{\alpha}_r^{(2)}|+|\bm{\alpha}_r^{(3)}|}\prod_{k=1}^{3}H_{\ell_k}(\bm{\alpha}_{l}^{(k)},\bm{\alpha}_{r}^{(k)}|\bm{\theta}^{(k)})\\\notag &\quad\times Z_p\left(\bm{\alpha}_r^{(1)}\cup\bm{\alpha}_l^{(2)}|\bm{\theta}^{(12)} \right)Z_p\left(\bm{\alpha}_r^{(2)}\cup\bm{\alpha}_l^{(3)}|\bm{\theta}^{(23)}\right)Z_p\left(\bm{\alpha}_r^{(3)}\cup\bm{\alpha}_l^{(1)}|\bm{\theta}^{(31)}\right)\period \end{align} Let us emphasize that our final expression~\eqref{generalformula} has a number of advantages. Firstly, the result is valid for the three-point functions built upon more general spin-chain vacua than the ones studied in the literature. Secondly, the result already demonstrates certain separation into the kinematical factor and the dynamical factor. Thirdly , the dynamical factor $\mathcal{D}_{\{\bm{\alpha}_{l,r}^{(1)},\bm{\alpha}_{l,r}^{(2)}\bm{\alpha}_{l,r}^{(3)}\}}$ is given essentially by a product of the pDWPF, each of which possesses determinant representation. One apparently unsatisfactory feature of~\eqref{generalformula} is that it still involves the sums over partitions, which become quite nontrivial especially when the number of magnons is large. As we shall show in the next subsection, however, for certain class of correlators the sum can be reduced to just a single term, by exploiting the \SU(2) symmetry.\footnote{A similar idea was utilized to simplify the three-point functions in the \SL(2) sector in~\cite{non-compact}.} This leads to a remarkably simple expression for which the semi-classical limit can be easily taken. ]]>

=& \left( \frac{1}{1+|z_1|^2}\right)^{\frac{\ell_1}{2} -M_1}\left( \frac{1}{1+|z_2|^2}\right)^{\frac{\ell_2}{2} -M_2}\left( \frac{1}{1+|z_3|^2}\right)^{\frac{\ell_3}{2}}\\ &\times z_{21}^{\ell_{12}-M_1-M_2}z_{32}^{\ell_{23}-M_2+M_1}z_{13}^{\ell_{31}-M_1+M_2}\mathcal{G}\comma\label{structuremixed} \end{aligned} } where the factor $\mathcal{G}$ stands for the term independent of $z_i$'s. As can be easily seen, the first line of~\eqref{structuremixed} coincides with the second line of~\eqref{generalformula}. On the other hand, the structure given in the second line of~\eqref{structuremixed} is not visible in the sum-over-partition expression~\eqref{generalformula}. In order to compare them more closely, let us expand both sides in powers of $z_3$. Upon this expansion, the second line of~\eqref{structuremixed} yields the following term as the highest-order term: \beq{ (-1)^{\ell_{31}-M_1+M_2}z_3^{\ell_3}\left( z_{21}^{\ell_{12}-M_1-M_2}\mathcal{G}\right)\period } On the other hand, if we expand each term in the sum in~\eqref{generalformula}, we obtain the following expression as the highest-order term:\footnote{Note that, since $\ket{\mathcal{O}_3}_L$ does not have any magnons, there is no sum over the partitions coming from $\mathcal{O}_3$.} \beq{ (-1)^{\ell_{31}-|\bm{\alpha}_l^{(1)}|}z_{3}^{\ell_3-|\bm{\alpha}_l^{(1)}|-|\bm{\alpha}_r^{(2)}|}\left( z_{21}^{\ell_{12}-|\bm{\alpha}_r^{(1)}|-|\bm{\alpha}_l^{(2)}|}\right)\mathcal{D}_{\{\bm{\alpha}_{l,r}^{(1)},\bm{\alpha}_{l,r}^{(2)}, \varnothing\}}\period } This shows that only a single term in the sum, for which $|\bm{\alpha}_l^{(1)}|=|\bm{\alpha}_r^{(2)}|=0$ holds, can produce the highest power $z_3^{\ell_3}$. Therefore, comparing the coefficients in front of $z_3^{\ell_3}$, we can determine $\mathcal{G}$ to be of the form \beq{ \begin{aligned} \mathcal{G}&=\left.(-1)^{-M_1+M_2}\mathcal{D}_{\{\bm{\alpha}_{l,r}^{(1)},\bm{\alpha}_{l,r}^{(2)}, \varnothing\}}\right|_{\bm{\alpha}_l^{(1)}=\bm{\alpha}_r^{(2)}=\varnothing}\\ &=(-1)^{M_2}\prod_{a=1}^{M_1}Q_{\bm{\theta}^{(31)}}^{+}(u_a^{(1)})\prod_{b=1}^{M_2}Q_{\bm{\theta}^{(23)}}^{-}(u_b^{(2)})Z_p\left( \bm{u}^{(1)}\cup \bm{u}^{(2)}|\bm{\theta}^{(12)}\right)\comma \end{aligned} } where the function $Q_{\bm{\theta}}(x)$ is defined by \beq{ Q_{\bm{\theta}}(x) \equiv \prod_{\theta \in \bm{\theta}}(x-\theta)\comma } and the superscripts $\pm$ denote the shift of the argument by $\pm i/2$. Let us now study the semi-classical limit of our three-point function. For this purpose, it is more convenient to introduce the ``rescaled" partial domain wall partition function\footnote{Note that it is the rescaled partial domain wall partition function, which has a simple semi-classical limit. In~\cite{Tailoring3}, it is called $\mathcal{A}$-functional.} defined by \beq{ \mathcal{Z}_p\left(\bm{u}^{(1)}\cup\bm{u}^{(2)}|\bm{\theta}^{(12)}\right) \equiv \frac{Z_p\left(\bm{u}^{(1)}\cup\bm{u}^{(2)}|\bm{\theta}^{(12)}\right)}{\prod_{x\in \bm{u}^{(1)}}Q_{\bm{\theta}^{(12)}}^{+}(x)\prod_{y\in \bm{u}^{(2)}}Q_{\bm{\theta}^{(12)}}^{-}(y)}\period } Then, $\big<\ket{\mathcal{O}_1}_L\comma\ket{\mathcal{O}_2}_L\comma \ket{\mathcal{O}_3}_L \big>$ takes the form \begin{align} \label{resultleft} \notag \big<\ket{\mathcal{O}_1}_L\comma \ket{\mathcal{O}_2}_L\comma \ket{\mathcal{O}_3}_L\big>\!=\!&\left( \frac{1}{1+|z_1|^2}\right)^{\frac{\ell_1}{2} -M_1}\left( \frac{1}{1+|z_2|^2}\right)^{\frac{\ell_2}{2} -M_2}\left( \frac{1}{1+|z_3|^2}\right)^{\frac{\ell_3}{2} }\\ &\times\!(z_1\!-\!z_2)^{\ell_{12}-M_1-M_2}(z_2\!-\!z_3)^{\ell_{23}-M_2+M_1}(z_3\!-\!z_1)^{\ell_{31}-M_{1}+M_2}\\ &\times\!\left(\prod_{a=1}^{M_1}Q^{+}_{\bm{\theta}^{(1)}}(u_a^{(1)})\prod_{b=1}^{M_2}Q^{-}_{\bm{\theta}^{(2)}}(u_b^{(2)})\mathcal{Z}_p\left(\bm{u}^{(1)}\cup\bm{u}^{(2)}|\bm{\theta}^{(12)}\right)\!\right),\notag \end{align} where we have neglected the factor $(-1)^{M_2}$ as it only changes the overall sign. Performing a similar analysis, we can also determine the contribution from the \SU(2)$_R$ spin chain and the result is given by \beq{ \begin{aligned} \big<\ket{\tilde{\mathcal{O}}_1}_R\comma \ket{\tilde{\mathcal{O}}_2}_R\comma \ket{\tilde{\mathcal{O}}_3}_R\big>=&\left( \frac{1}{1+|\tilde{z}_1|^2}\right)^{\frac{\ell_1}{2} }\left( \frac{1}{1+|\tilde{z}_2|^2}\right)^{\frac{\ell_2}{2}}\left( \frac{1}{1+|\tilde{z}_3|^2}\right)^{\frac{\ell_3}{2} -\tilde{M}_3}\\ &\times(\tilde{z}_1-\tilde{z}_2)^{\ell_{12}+\tilde{M}_3}(\tilde{z}_2-\tilde{z}_3)^{\ell_{23}-\tilde{M}_3}(\tilde{z}_3-\tilde{z}_1)^{\ell_{31}-\tilde{M}_{3}}\\ &\times\left(\prod_{a=1}^{M_3}Q^{+}_{\tilde{\bm{\theta}}^{(3)}}(\tilde{u}_a^{(3)})\mathcal{Z}_p\left( \tilde{\bm{u}}^{(3)}\cup\varnothing|\tilde{\bm{\theta}}^{(31)}\right)\right)\period \end{aligned}\label{resultright} } Note that, since the result is completely factorized into the \SU(2)$_L$ and the \SU(2)$_R$ parts, we can introduce independent sets of the inhomogeneities for the \SU(2)$_R$ sector denoted by $\tilde{\theta}$'s. The tree-level structure constant can then be obtained by setting $\theta$'s and $\tilde{\theta}$'s to zero. Now the semi-classical limit of the three-point coupling constant can also be easily studied using the results of~\cite{Tailoring3,Kostov1,Kostov2} and we obtain, up to a phase, \beq{ \begin{aligned} C_{123}&=\frac{\sqrt{\ell_1\ell_2\ell_3}}{N_c}k_L \, k_R\, c_{123}\comma\\ \log c_{123} &\sim \oint_{\mathcal{C}_{\bm{u}^{(1)}}\cup \mathcal{C}_{\bm{u}^{(2)}}}\frac{du}{2\pi i} {\rm Li}_2 \left(e^{ip_1+ip_2 +i\ell_3/2u} \right)+\oint_{\mathcal{C}_{\tilde{\bm{u}}^{(3)}}}\frac{du}{2\pi i} {\rm Li}_2 \left(e^{ip_3 +i(\ell_2-\ell_1)/2u} \right)\\ &\quad-\frac{1}{2}\oint_{\mathcal{C}_{\bm{u}^{(1)}}}\frac{du}{2\pi} {\rm Li}_2 \left( e^{2i p_1}\right)\!-\!\frac{1}{2}\oint_{\mathcal{C}_{\bm{u}^{(2)}}}\frac{du}{2\pi} {\rm Li}_2 \left( e^{2i p_2}\right)\!-\!\frac{1}{2}\oint_{\mathcal{C}_{\tilde{\bm{u}}^{(3)}}}\frac{du}{2\pi} {\rm Li}_2 \left( e^{2i \tilde{p}_3}\right)\period \end{aligned} } Here $k_L$ and $k_R$ are kinematical factors given by the first two lines on the left hand side of~\eqref{resultleft} and~\eqref{resultright} respectively, $p_{n}(u)$ and $\tilde{p}_{n} (u)$ are the quasi-momenta given by \beq{ p_{n}(u) = \sum_{v\in \bm{u}^{(n)}} \frac{1}{u-v}- \frac{\ell_n}{2u}\comma\qquad \tilde{p}_{n}(u) = \sum_{v\in \tilde{\bm{u}}^{(n)}} \frac{1}{u-v}- \frac{\ell_n}{2u}\comma } and the integration contours $\mathcal{C}_{\bm{u}^{(n)}}$ and $\mathcal{C}_{\tilde{\bm{u}}^{(n)}}$ encircle\footnote{As briefly discussed in~\cite{Kostov2}, the contours are in general complicated and the case-by-case analysis is necessary.} the Bethe roots $\bm{u}^{(n)}$ and $\tilde{\bm{u}}^{(n)}$ respectively. So far, we have seen that the mixed correlators have simple expressions, which allow us to study the semi-classical limit with ease. The remaining class of three-point functions are the ones for which all the three operators are of the same type. We call such three-point functions ``unmixed''. It turns out that, in the case of the unmixed correlators, several different terms in the sum in~\eqref{generalformula} contribute to the the highest power of $z_i$'s, and therefore the result cannot be simplified by the straightforward application of the aforementioned logic. In addition, the prediction from the semi-classical computation based on the coherent states (to be reported in~\cite{KKN3}) does not take a form which can be readily obtained from the pDWPF. These two observations indicate that the unmixed correlators are much more complicated objects. Nevertheless, studying such three-point functions is important for the following reason: the pDWPF is the quantity which describes the skew-symmetric product of two spin-chain state. Therefore, the fact that the mixed correlators can be reduced to the pDWPF suggests that such three-point functions are characterized essentially by the integrability governing the two-point function, which is already fairly well-understood. This in turn means that, in order to reveal the genuine ``integrability for the three-point functions'', we do need to study the unmixed correlators, which cannot be simplified into the pDWPF. ]]>

\comma\label{omega2omega2} } where $i$, $j$ and $k$ are the indices for the auxiliary space, and $\Omega_n $ and $\overleftarrow{\Omega}_n $ are the monodromy and the ``reverse-ordered'' monodromy\footnote{Note that, owing to the relation~\eqref{crossed2}, the reverse-ordered monodromy is equivalent to the monodromy which appeared in~\eqref{eq-37}: $\overleftarrow{\Omega}_n(-u)=(-1)^{\ell_n}\mathcal{C}\circ \Omega_n(u)$.} for the operator $\mathcal{O}_n$, defined by \begin{align} \Omega_n(u+i/2)&\equiv {\rm L}^{(n)}_{1}(u-\theta^{(n)}_{1}+i/2)\cdots{\rm L}^{(n)}_{\ell_n}(u-\theta^{(n)}_{\ell_n}+i/2 )\comma\label{omegan}\\ \overleftarrow{\Omega}_n(-u+i/2)&\equiv {\rm L}^{(n)}_{\ell_n}(\theta^{(n)}_{\ell_n}-u+i/2 )\cdots{\rm L}^{(n)}_{1}(\theta^{(n)}_{1}-u+i/2)\period\label{revomegan} \end{align} Here ${\rm L}^{(n)}_k$ and $\theta^{(n)}_k$ respectively denote the Lax operator and the inhomogeneity parameter for the $k$-th site of the spin-chain state $\ket{\mathcal{O}_n}_{L}$, and $\ell_n$ is the length of the operator $\mathcal{O}_n$. Here again the inhomogeneities are identified as $\theta^{(1)}_k=\theta^{(2)}_{\ell-k+1}$, as discussed already in section~\ref{subsec:cutandsew}. Using the unitarity relation~\eqref{unitarity} repeatedly, we can show that~\eqref{omega2omega2} is proportional to the skew-symmetric product without monodromy insertions, which is depicted in the figure (b) of figure~\ref{2pt_monod}: \beq{ \eqref{omega2omega2}=\delta_{ik}(-1)^{\ell} f_{12}(u)\big< \ket{\mathcal{O}_1}_L\comma \ket{\mathcal{O}_2}_L \big>\comma\label{noomega} } where the prefactor $f_{12}(u)$ is given by \beq{ &f_{12}(u)\equiv\prod_{k=1}^{\ell}\left((u-\theta^{(1)}_k)^2+1\right) =\prod_{k=1}^{\ell}\left((u-\theta^{(2)}_k)^2+1\right)\period } Let us next apply the crossing relation to each Lax operator constituting $\overleftarrow{\Omega}_2$ in~\eqref{omega2omega2}. Since the $k$-th site of the operator $\mathcal{O}_2$ is contracted with the $(\ell-k+1)$-th site of the operator $\mathcal{O}_1$, the Lax operator transforms under the application of the crossing relation as \beq{ {\rm L}^{(2)}_{k}(-u+\theta_k^{(2)})\to -{\rm L}^{(1)}_{\ell-k+1}(u-\theta_{\ell-k+1}^{(1)})\comma } where we used the identifications of the inhomogeneity parameters~\eqref{inhomoeq}. Thus, after the successive application of the crossing relation, we arrive at the following expression, which is depicted in the figure of figure~\ref{2pt_monod}: \beq{ \eqref{omega2omega2}=(-1)^{\ell}\big<\Big( \Omega_1^{-}(u)\Big)_{ij} \ket{\mathcal{O}_1}_L\comma \Big( \Omega_2^{+}(u)\Big)_{jk} \ket{\mathcal{O}_2}_L \big>\period\label{omega+omega-} } The superscripts $\pm$ on the monodromy operator denotes the shift of the argument $ \Omega^{\pm}(u)\equiv \Omega (u\pm i/2)$. Then, by equating the right hand sides of~\eqref{noomega} and~\eqref{omega+omega-}, we obtain the monodromy relation for the two-point function: \beq{ \big<\Big( \Omega_1^{-}(u)\Big)_{ij} \ket{\mathcal{O}_1}_L\comma \Big( \Omega_2^{+}(u)\Big)_{jk} \ket{\mathcal{O}_2}_L \big>=\delta_{ik}\,f_{12}(u)\big<\ket{\mathcal{O}_1}_L\comma\ket{\mathcal{O}_2}_L \big> \period\label{monod2_L} } One can write down a similar relation also for the \SU(2)$_R$ chain as \beq{ \big<\Big( \tilde{\Omega}_1^{-}(u)\Big)_{ij} \ket{\tilde{\mathcal{O}}_1}_R\comma \Big( \tilde{\Omega}_2^{+}(u)\Big)_{jk} \ket{\tilde{\mathcal{O}}_2}_R \big>=\delta_{ik}\,\tilde{f}_{12}(u)\big<\ket{\tilde{\mathcal{O}}_1}_R\comma\ket{\tilde{\mathcal{O}}_2}_R \big> \comma\label{monod2_R} } where $\tilde{\Omega}_n(u)$ are the monodromy matrices for the \SU(2)$_R$ chain and $\tilde{f}_{12}(u)$ is given in terms of the inhomogeneity for the \SU(2)$_R$ chain $\tilde{\theta}^{(n)}_k$ by \beq{ \tilde{f}_{12}(u)\equiv \prod_{k=1}^{\ell}\left((u-\tilde{\theta}_k^{(1)})^2+1 \right)=\prod_{k=1}^{\ell}\left((u-\tilde{\theta}_k^{(2)})^2+1 \right)\period } The monodromy relations~\eqref{monod2_L} and~\eqref{monod2_R} are the embodiment of the integrability for the two-point function. As the two-point function is determined by the spectrum of the operators, they should be essentially equivalent to the integrable structures already known in the spectral problem. However, it might be interesting to clarify the relation with the conventional formalism and ask if these new formalism helps to deepen the understanding of the spectral problem. ]]>

\\ &=\delta_{il}f_{123}(u)\big<\ket{\mathcal{O}_1}_L\comma\ket{\mathcal{O}_2}_L\comma \ket{\mathcal{O}_3}_L\big>\comma \end{aligned}\label{monod3_L} } where $f_{123}(u)$ is defined by\footnote{For a definition of $\ell_{ij}$, see~\eqref{eq-20}.} \beq{ f_{123}(u)\equiv \prod_{i=1}^{\ell_{31}}\left((u-\theta_i^{(1)})^2+1\right) \prod_{j=1}^{\ell_{12}}\left((u-\theta_j^{(2)})^2+1\right) \prod_{k=1}^{\ell_{23}}\left((u-\theta_k^{(3)})^2+1\right)\comma } and $\Omega^{+|-}_2(u)$ denotes a product of the monodromy matrices on the left and the right sub-chains of $\mathcal{O}_2$ whose arguments are shifted by $+i/2$ and $-i/2$ respectively. More specifically, the relevant monodromy matrices are given by \beq{ \begin{aligned} \Omega^{-}_1(u)&={\rm L}^{-}_{1}(u-\theta^{(1)}_{1})\cdots {\rm L}^{-}_{\ell_{31}}(u-\theta^{(1)}_{\ell_{31}}){\rm L}^{-}_{\ell_{31}+1}(u-\theta^{(1)}_{\ell_{31}+1})\cdots {\rm L}^{-}_{\ell_1}(u-\theta^{(1)}_{\ell_1})\comma\\ \Omega^{+|-}_2(u)&={\rm L}^{+}_{1}(u-\theta^{(2)}_{1})\cdots {\rm L}^{+}_{\ell_{12}}(u-\theta^{(2)}_{\ell_{12}}){\rm L}^{-}_{\ell_{12}+1}(u-\theta^{(2)}_{\ell_{12}+1})\cdots {\rm L}^{-}_{\ell_2}(u-\theta^{(2)}_{\ell_2})\comma\\ \Omega^{+}_3(u)&={\rm L}^{+}_{1}(u-\theta^{(3)}_{1})\cdots {\rm L}^{+}_{\ell_{23}}(u-\theta^{(3)}_{\ell_{23}}){\rm L}^{+}_{\ell_{23}+1}(u-\theta^{(3)}_{\ell_{23}+1})\cdots {\rm L}^{+}_{\ell_3}(u-\theta^{(3)}_{\ell_3})\period \end{aligned}\label{defshift} } For the \SU(2)$_R$ chain, the corresponding form of the monodromy relation can be written~as \beq{ \begin{aligned} &\big<\Big(\tilde{\Omega}^{-}_1(u)\Big)_{ij}\ket{\tilde{\mathcal{O}}_1}_R\comma\Big(\tilde{\Omega}_2^{+|-}(u)\Big)_{jk}\ket{\tilde{\mathcal{O}}_2}_R \comma \Big(\tilde{\Omega}_3^{+}(u)\Big)_{kl}\ket{\tilde{\mathcal{O}}_3}_R\big>\\ &=\delta_{il}\tilde{f}_{123}(u)\big<\ket{\tilde{\mathcal{O}}_1}_R\comma\ket{\tilde{\mathcal{O}}_2}_R\comma \ket{\tilde{\mathcal{O}}_3}_R\big>\comma \end{aligned}\label{monod3_R} } where $\tilde{f}_{123}(u)$ is defined by \beq{ &\tilde{f}_{123}(u)\equiv \prod_{i=1}^{\ell_{31}}\left((u-\tilde{\theta}_i^{(1)})^2+1\right) \prod_{j=1}^{\ell_{12}}\left((u-\tilde{\theta}_j^{(2)})^2+1\right) \prod_{k=1}^{\ell_{23}}\left((u-\tilde{\theta}_k^{(3)})^2+1\right)\period } As in the \SU(2)$_L$ sector, $\tilde{\Omega}^{+|-}_2(u)$ denotes a product of the monodromy matrices on the left and the right sub-chains whose arguments are shifted by $+i/2$ and $-i/2$ respectively. \begin{figure} \centering \includegraphics[clip,height=3.5cm]{3pt_monod-m.png} ]]>

\!+\! \big<\ket{\mathcal{O}_1}_L\comma S_{\ast}\ket{\mathcal{O}_2}_L\comma \ket{\mathcal{O}_3}_L\big> \!+\! \big<\ket{\mathcal{O}_1}_L\comma \ket{\mathcal{O}_2}_L\comma S_{\ast}\ket{\mathcal{O}_3}_L\big> \!=\! 0\comma \label{ward} } where $S_{\ast}$ are the global \SU(2) generators and $\ast$ stands for $1$, $2$ or $3$. These global Ward identities are quite useful in fixing the kinematical dependence of the three-point functions, as described in the appendix~\ref{ap}. Naturally it would be quite interesting and important to study the non-trivial relations obtained at the sub-leading levels and see if we can exploit them to understand the structure of the three-point functions.\footnote{At the sub-leading order,~\eqref{monod3_L} produces a set of non-trivial identities involving operators which act non-locally on the spin chains. These identities can be regarded as a sort of Yangian invariance for the three-point functions. Simlar relations are discussed in the context of the scattering amplitudes in~\cite{spectralreg1,spectralreg2,spectralreg3,spectralreg4,Chicherin1,Chicherin2} and it would be interesting to clarify the connection.} \fussy As for the importance of the monodromy relation, we already have a supporting evidence from the strong coupling computation performed in~\cite{KK3}. In that analysis the three-point function in the \SU(2) sector was determined from the following relation for the monodromy matrices defined on the classical string world-sheet: \beq{ \Omega_1(x)\Omega_2(x)\Omega_3(x)=\bm{1}\period\label{om1om2om3} } This relation, which is a direct consequence of the classical integrability of the string sigma model, is a clear manifestation of the integrability for the three-point function at strong coupling and was indeed an essential ingriedient in the computation of the three-point functions. The relations we derived here,~\eqref{monod3_L} and~\eqref{monod3_R}, can be regarded as the weak coupling counter-part of~\eqref{om1om2om3} and its generalization. The similarity becomes more apparent if we take the so-called semi-classical limit of the spin chain, in which the length of the chain and the number of the magnons are both large. To study the low energy excitation in this limit, we need to use the rescaled spectral parameter $u=\ell u^{\prime}$, and send $\ell$ to $\infty$ keeping $u^{\prime}$ finite. In terms of this rescaled parameter, the shifts of the spectral parameter in $\Omega_n^{\pm}$, $\Omega_2^{+|-}$ and so on become negligible. Furthermore, in this limit, the three-point function will be well-approximated by coherent states. Then the monodromy matrices, which are originally quantum operators acting on the spin chains, become classical. Therefore, in such a limit the relations~\eqref{monod3_L} and~\eqref{monod3_R} exactly take the same form as~\eqref{om1om2om3}. As will be discussed in the forthcoming publication~\cite{KKN3}, we can use~\eqref{monod3_L} and~\eqref{monod3_R} to directly study the semi-classical behavior of the three-point functions at weak coupling without relying on the explicit determinantal expressions for the scalar products of the XXX spin chain. ]]>

=0 \period \end{align} It is evident that this has exactly the same form as the global conformal Ward identity for three-point functions in 2d CFT if we identify $-L_i$ with the conformal dimensions. Thus, the $z_i$ dependence can be uniquely fixed~\cite{BPZ} as \begin{align} \big< |\hat{\mathcal{O}}_1 \rangle_L,|\hat{\mathcal{O}}_2 \rangle_L,|\hat{\mathcal{O}}_3 \rangle_L \big> \propto z_{21}^{L_{12}}z_{32}^{L_{23}}z_{13}^{L_{31}}\comma \end{align} where $z_{ij}\equiv z_i-z_j$ and $L_{ij}\equiv L_i+L_j-L_k$. Therefore, the kinematical dependence of the three point function for the \SU(2)$_L$ sector is given by the simple form \beq{ \begin{aligned} \big< |\mathcal{O}_1 \rangle_L,|\mathcal{O}_2 \rangle_L,|\mathcal{O}_3 \rangle_L\big> \propto& \left( \frac{1}{1+|z_1|^2} \right)^{L_1} \left( \frac{1}{1+|z_2|^2} \right)^{L_2} \left( \frac{1}{1+|z_3|^2} \right)^{L_3}\\ &\times z_{21}^{L_{12}}z_{32}^{L_{23}}z_{13}^{L_{31}}\period \end{aligned}\label{kineL} } Similarly, for the \SU(2)$_R$ sector the result is \beq{ \begin{aligned} \big< |\tilde{\mathcal{O}}_1 \rangle_R,|\tilde{\mathcal{O}}_2 \rangle_R,|\tilde{\mathcal{O}}_3 \rangle_R\big> \propto& \left( \frac{1}{1+|\tilde{z}_1|^2} \right)^{R_1} \left( \frac{1}{1+|\tilde{z}_2|^2} \right)^{R_2} \left( \frac{1}{1+|\tilde{z}_3|^2} \right)^{R_3}\\ &\times \tilde{z}_{21}^{R_{12}}\tilde{z}_{32}^{R_{23}}\tilde{z}_{13}^{R_{31}}\comma \end{aligned}\label{kineR} } where $R_i$ is given by $\ell_i/2 -\tilde{M}_i$. It is important to note that the relations~\eqref{kineL} and~\eqref{kineR} take the following form in terms of the polarization spinors, \beq{ \begin{aligned} \big< |\mathcal{O}_1 \rangle_L,|\mathcal{O}_2 \rangle_L,|\mathcal{O}_3 \rangle_L\big> &\propto\langle\mathfrak{n}_1,\mathfrak{n}_2 \rangle^{L_{12}}\langle \mathfrak{n}_2,\mathfrak{n}_3 \rangle^{L_{23}}\langle \mathfrak{n}_3,\mathfrak{n}_1 \rangle^{L_{31}} \comma\\ \big< |\tilde{\mathcal{O}}_1 \rangle_R,|\tilde{\mathcal{O}}_2 \rangle_R,|\tilde{\mathcal{O}}_3 \rangle_R\big> &\propto\langle\tilde{\mathfrak{n}}_1,\tilde{\mathfrak{n}}_2 \rangle^{R_{12}}\langle \tilde{\mathfrak{n}}_2,\tilde{\mathfrak{n}}_3 \rangle^{R_{23}}\langle \tilde{\mathfrak{n}}_3,\tilde{\mathfrak{n}}_1 \rangle^{R_{31}}\comma \end{aligned} } where $\langle\mathfrak{n},\mathfrak{m} \rangle\equiv \det \left(\mathfrak{n},\mathfrak{m} \right)$. This is precisely the structures observed in the computation at strong coupling~\cite{KK3}. It should be useful to make a small remark on the uniqueness of the kinematical dependence as determined by the symmetry argument. Although the results~\eqref{kineL} and~\eqref{kineR} above for the ``\SU(2) sector'' are unique, this is not true in the case of higher rank sectors. For instance, in the \SO(6) sector, the symmetry argument alone cannot fix the dependence completely and there exist several possible R-symmetry tensorial structures. In such cases, the three-point function is given by a linear combination of such allowed structures, whose coefficients depend on dynamics, for instance on 't Hooft coupling. Indeed, for the \SO(2,4) sector, the existence of a large number of tensorial structures was found in~\cite{Spinning}. ]]>