text:
\section{Introduction}
One of the main reasons for trying to solve a gauge theory such as $\mathcal{N}=4$ SYM is to search for hidden structures pointing towards alternative descriptions of Quantum Field Theory. { As a byproduct, we also hope to develop new techniques that will allow us to do computations which are currently beyond reach and which will improve our understanding of String Theory and Quantum Gravity.}

To illustrate how such structures are often hidden we will consider the problem of computing correlation functions of local gauge invariant operators in the planar limit of $\mathcal{N}=4$ SYM.
For simplicity, let us consider operators made out of two complex scalars only, say $Z$ and $X$. As usual we identify them as spins for notational convenience,
\begin{equation}
{\rm Tr} (ZX\dots) \leftrightarrow \left|\uparrow\downarrow\dots\right\> \,.
\end{equation}
We use $N$ for the number of $X$ fields and $L$ for the total number of fields. Then, the gauge invariant operators with definite total dimension $\Delta=L+\gamma$ are the eigenvectors of the
dilatation operator which can be written as a spin chain Hamiltonian as \cite{MZ,BKS}
\begin{equation}
H=\sum_{n=1}^L \[ (2g^2-8g^4) \, \mathbb{H}_{n,n+1} +2 g^4\,  \mathbb{H}_{n,n+2}  + \mathcal{O}(g^6)\] \la{dilatation0}\;.
\end{equation}
Here $\mathbb{H}_{nm}=\mathbb{I}_{nm}-\mathbb{P}_{nm}$, with $\mathbb{I}$, $\mathbb{P}$ the identity and permutation operator, respectively. The energies of
this Hamiltonian are precisely the anomalous part of the total dimension, $\gamma$.
We can already identify remarkable simplifications which come about when considering physical quantities but which are very much obscured in the intermediate steps:
take for example the dilatation operator (\ref{dilatation0}). If we were to write its next few loop orders it would require pages and pages with no obvious structure present. Nevertheless, its eigenvalues, which are the physical quantities $\gamma$  can be obtained very neatly as we now briefly review. We can think of the states diagonalizing (\ref{dilatation0}) as $N$ excitations $\downarrow$ moving with momenta ${\bf p}=\{p_1,\dots,p_N\}$ on top of a ferromagnetic sea of $\uparrow$ fields. The scattering of two magnons is governed by the magnon S-matrix $S(p_i,p_j)$.\footnote{{ The scattering of $N$ particles is also governed by the two magnon S-matrix since
the many body S-matrix factorizes into a product of two body scattering processes in an Integrable theory.}} The quantization of these momenta is given by Bethe equations\footnote{See section \ref{contactSec} for a very brief review and \cite{paper1} for a more extended one.}
\begin{equation}
L p_k + \sum_{l \neq k} \frac{1}{i} \log S(p_l,p_k) = 2\pi n_k \,,\qquad n_k \in \mathbb{Z} \,. \la{BetheEq0}
\end{equation}
The S-matrix is known to all loop orders \cite{Beisert,BES} and is quite simple. Once the momenta are fixed the energy of the state is obtained by summing the energies $\epsilon(p_j)$ of each individual excitation,
\begin{equation}
\gamma({\bf p})= \sum_{j=1}^N \epsilon(p_j)\,, \qquad \epsilon(p)=8g^2 \sin^2(p/2)-32 g^4\sin^4(p/2)+\mathcal{O}(g^6) \la{gamma0} \,.
\end{equation}
This scattering picture needs to be corrected if the chains are too small \cite{Matthias,TBA}. That is, the Bethe ansatz (\ref{BetheEq0}) is asymptotic.  Equations (\ref{gamma0}) and (\ref{BetheEq0}) are a striking example of how beauty emerges at the end of the day for physical quantities.

Once the dimensions of operators are known we know their two point functions. The next natural observables are the structure constants which govern the three point functions. For three point functions we also need the eigenvectors of the dilation operator.
 After all we need to Wick contract them (at leading order) and then decorate this contraction by loops. However, the eigenvectors can be
 horrendous at first sight.
 More specifically, at two loops (and higher) there will be some local corrections to the wave functions, called \textit{contact terms}, when two or more magnons are close together. When four excitations are close together the corresponding two loop contact term correction is as messy as to occupy one full page!, see appendix \ref{C4}.

For the spectrum problem we could avoid dealing with these contact terms by considering asymptotically large chains where there is always enough
room for the excitations to move freely. Then, to quantize the momenta of the excitations all we care about is the S-matrix and to compute their energy we can consider a region where they are well separated. However, when we move to three point functions
the contact terms come into play. When we contract the three operators we need to know the exact wave function. All regions will be probed and all of them will contribute. On the other hand,
the structure constants $C_{123}$ are physical quantities and as such, there ought to be some nice description where all the complications due to the contact terms and the loop corrections conspire together to give a neat final result.

Confirming this expectation is the main goal of the current paper. To arrive at this result we propose an algebraic description of the eigenvectors of the dilation operator (\ref{dilatation0}) which automatically incorporates all possible contact terms. This approach is based on the so called $\Theta$--morphism which gives the name to the paper.

Let us end this introduction by mentioning that there are numerous other exciting hints at hidden structures, other than the $\Theta$--morphism \cite{BDS,boost,Hubbard,sigma,didina}. They strongly inspired this work.
It would be wonderful to explore in detail the different connections and differences between all these approaches.

This paper is organized as follows. In section \ref{setup} we translate the problem of computing structure constants into the spin chain language. The rest of the paper is divided into two main parts. In section \ref{coorSec} we follow the coordinate Bethe ansatz approach combined with some inspired guesses based on our experience with the spectrum problem. This section will contain several conjectures for the one loop correction to the structure constants and also for some all loop generalizations. In sections \ref{algSec} and \ref{secMor1} we follow an algebraic Bethe ansatz approach and we develop the $\Theta$--morphism map. This will allow us to derive the most general $SU(2)$ structure constants and prove all the one loop conjectures of section \ref{coorSec}, based on the coordinate Bethe ansatz approach.  We conclude in section \ref{Conc}. Several appendices contain complementary material to the main text.

\section{Structure Constants and Quantum Spin Chains} \la{setup}
In this paper we will consider structure constants in the $SU(2)$ setup introduced in \cite{paper1}.
Let us recall how these objects can be efficiently computed using the spin chain picture. At tree level all we need to do is to Wick contract the three operators. These operators are the eigenvectors of the dilatation operator (\ref{dilatation0}).
At one loop level (a) we need to Wick contract the operators which are the eigenvectors of the two loop dilatation operator  and (b) we need to include Hamiltonian insertions coming from Feynman diagrams
\cite{C123p1,C123p2,C123p3}
(see figure \ref{figure}). In total
\begin{figure}[t]
\begin{center}
\includegraphics[width=140mm]{HIns2.pdf}
\end{center}
\caption{Internal loop diagrams results in the insertions of the Hamiltonian densities at $6$ junction points \cite{C123p1,C123p2,C123p3}. This leads to (\ref{C123}) and (\ref{tocompute}) in the spin chain language.} \la{figure}
\end{figure}
\beqa
C_{123}&=&\verb"norms" \times \verb"simple" \times \verb"involved" +\mathcal{O}(g^4) \la{C123}
\eeqa
where
\beqa
\verb"norms" &=&{\sqrt{L_1L_2L_3}}/{\sqrt{\<{\bf 1}|{\bf 1}\>\<{\bf 2}|{\bf 2}\>\<{\bf 3}|{\bf 3}\>}} \,, \nonumber \\
\verb"simple" &=&\<\underbrace{\downarrow\dots\downarrow}_{N_3} \underbrace{\uparrow\dots\uparrow}_{L_3-N_3}|1-g^2 \mathbb{H}_{N_{3},N_{3}+1}-g^2 \mathbb{H}_{L_3,1} |{\bf 3}\> \la{tocompute}\,,\\
\verb"involved" &=&\<{\bf 1}|1-g^2 \mathbb{H}_{L_{12},L_{12}+1}-g^2 \mathbb{H}_{L_1,1} \nonumber
|
{{i_1}\dots{i_{L_{12}}}}
\underbrace{\downarrow\dots\downarrow}_{N_3}\>\\
\nonumber&&\<
{{i_1}\dots {i_{L_{12}}}}
\underbrace{\uparrow\dots\uparrow}_{L_3-N_3}|
1-g^2 \mathbb{H}_{L_{12},L_{12}+1}-g^2 \mathbb{H}_{L_2,1}|{\bf 2}\>\,.
\eeqa
Where $i_a$ can be $\uparrow$ or $\downarrow$ and one sums over all possible $2^{L_{12}}$ intermediate states.
Here $|\bf a\>$ stands for the corrected two loop Bethe eigenstate corresponding to the operator $\mathcal{O}_a$. The length and number of excitations of this operator are denoted by $L_a$ and $N_a$ respectively.
Of course, for these expressions to make sense and to be non zero we need
\begin{equation}
N_1 = N_2 + N_3 \qquad \text{and} \qquad L_{12} = L_1-N_3=L_2-L_3+N_3 \la{constraint}
\end{equation}
where $L_{12}$ is the number of contractions between operators ${\mathcal O}_1$ and ${\mathcal O}_2$. For any pair of operators ${\mathcal O}_i$ and ${\mathcal O}_j$ we have $2L_{ij}=L_i+L_j-L_k$ where $k\neq i,j$.

The computation of structure constants in $\mathcal{N}=4$ SYM in the $SU(2)$ sector up to one loop order is thus mapped into a very precise spin chain problem:
we simply need to be able to compute all spin chain scalar products and expectation values appearing in (\ref{tocompute}). At order $g^0$ this was solved in \cite{paper1,omar}. One of the main goals of the current paper is to compute the first quantum correction, of order $g^2$.
This would be the first step toward an all loop asymptotic expression for the structure constants.
Obviously we want to find reasonable conjectures for all loop results based on the tree level and one loop results
based on our experience with the spectrum problem.
As an example, we will now consider the first contribution in (\ref{C123}), the $\verb"norms"$. For that,  we first need to work out the precise form of the two loop eigenvectors $|{\bf a}\>$ which will allow us to ask nonambiguous questions about their norms.

\section{Two Loop States and Contact Terms} \la{contactSec}
The dilatation operator in $\mathcal{N}=4$ SYM can be thought of as a long-range spin chain Hamiltonian whose interaction range coincides with the order of perturbation theory. At two loops we have (\ref{dilatation0}).
The single magnon state is given by
\begin{equation}
|p\> =\sum_{n=1}^L e^{ipn} |n\> \,, \qquad  |n_1,n_2,\dots \>= |\overbrace{\underbrace{\uparrow \dots \uparrow}_{n_1-1}\! \downarrow \uparrow \dots\uparrow}^{n_2-1}\! \downarrow \uparrow \dots \> = \sigma_-^{(n_1)}\sigma_-^{(n_2)}\dots \left|\uparrow\dots \uparrow\right\>\;.
\end{equation}
This follows from translation invariance and is true to all orders in perturbation theory. The two magnon state is more interesting. At two loops we have
\begin{equation}
|p_1,p_2\> =\sum_{n_1 < n_2} \(e^{ip_1n_1+ip_2n_2}+e^{ip_2n_1+ip_1n_2} S(p_2,p_1) \)\(1+ \delta_{n_2,n_1+1} \,g^2\, \mathbb{C}_{\bullet\bullet}^{(2)}(p_2,p_1) \) |n_1,n_2\> \la{2magnon}\;.
\end{equation}
The physical picture is clear: we have an incoming wave ($e^{ip_1n_1+ip_2n_2}$) describing two particles with momentum $p_1$ and $p_2$. They can then collide. In $1+1$ dimensions when two particles collide their momentum can at most be exchanged. This is a consequence of conservation of both energy and momentum. The outgoing plane wave ($e^{ip_2n_1+ip_1n_2}$) describes the particles with their momenta exchanged. The relative coefficient between the incoming and outgoing plane waves is what we define as the S-matrix, which turns out to be given by \cite{BDS,Matthias,BES}
\begin{equation}
S(p,k)= \frac{u(k)-u(p)-i}{u(p)-u(k)+i} \,, \qquad u(p)=\frac{1}{2}\cot\(\frac{p}{2}\) \(1+ 8g^2 \sin^2 \frac{p}{2} + \dots \) \la{Smatrix}\;.
\end{equation}
This is all we have as long as the particles are well separated. When they are within the interaction range of the Hamiltonian the two magnons will be in a complicated interacting state. This is taken into account by introducing the so called \textit{fudge factors} \cite{Matthias} which we also denote as \textit{contact terms}. The purpose of this section is to study these terms. At two loops, and for two magnons, we only need
\begin{equation}\mathbb{C}_{\bullet\bullet}^{(2)}(p_2,p_1) = -4 \sin(p_1/2) \sin(p_2/2)\sec((p_1+p_2)/2)  \la{C2}\,.\end{equation}
The superscript indicates that this is a two loop effect while the subscript indicates that the two magnons need to be close to each other.

The higher loop picture should be clear; new fudge factors have to be added with longer and longer range. For example, at three loops, the two magnon the state (\ref{2magnon}) is modified by the simple replacement of $\(1+ \delta_{n_1+1,n_2}\,g^2\, \mathbb{C}_{\bullet\bullet}^{(2)}(p_2,p_1) \) $ by
\begin{equation}
 \Big(1+ \delta_{n_2,n_1+1} \,\[g^2\, \mathbb{C}_{\bullet\bullet}^{(2)}(p_2,p_1) +g^4\, \mathbb{C}_{\bullet\bullet}^{(3)}(p_2,p_1)\]+ \delta_{n_2,n_1+2} \,g^4\, \mathbb{C}_{\bullet\circ\bullet}^{(3)}(p_2,p_1) \Big) \,.
\end{equation}
Also, the S-matrix needs to be corrected further and at four loop order a new contribution to the S-matrix -- the so called BES dressing phase \cite{BES} -- shows up. For now let us continue our two loop discussion but move to a higher number of excitations.

Interesting new structures arise when we move to three (and more) magnons. For three magnons we have
\begin{equation}|p_1,p_2,p_3\> =\sum_{n_1 < n_2<n_3} \!\!\!\![\phi_{123}+\phi_{213} S_{21}+\phi_{132} S_{32} +\phi_{312} S_{31}  S_{32} +\phi_{231}S_{31} S_{21}+\phi_{321} S_{32}  S_{31}  S_{21}] |n_1,n_2,n_3\>\nonumber\end{equation} where $S_{ab}=S(p_a,p_b)$ and
\begin{equation} \phi_{abc}= \exp( ip_a n_1+i p_b n_2+ i p_c n_3) (1+g^2 \delta_{abc})\,.\end{equation}
Finally -- and most relevant for the current discussion -- $\delta_{abc}$ are the contact terms. We have
\begin{equation}
\delta_{abc} =  \left\{ \begin{array}{ll} 0 & \text{for  } n_2-n_1>1 \text{ and } n_3-n_2>1  \\
 \mathbb{C}_{\bullet\bullet}^{(2)}(p_a,p_b) &   \text{for  } n_2-n_1=1 \text{ and } n_3-n_2>1 \\
 \mathbb{C}_{\bullet\bullet}^{(2)}(p_b,p_c) &   \text{for  } n_2-n_1>1 \text{ and } n_3-n_2=1 \\
 \mathbb{C}_{\bullet\bullet\bullet}^{(2)}(p_a,p_b,p_c) & \text{for  } n_2-n_1=1 \text{ and } n_3-n_2=1 \\
 \end{array} \right.
\end{equation}
We see that on top of the two magnon contact term we now need a new contact term for the case when the three magnons are all close to each other.
States with more magnons follow the same pattern. For $N$ magnons we will have $N!$  terms in the wave function and $N-1$ different contact terms: $\mathbb{C}^{(2)}_{\bullet\bullet}(p_1,p_2), \mathbb{C}^{(2)}_{\bullet\bullet\bullet}(p_1,p_2,p_3), \dots, \mathbb{C}^{(2)}_{\bullet\dots\bullet}(p_1,\dots,p_N)$.
By explicit diagonalization of the two loop dilatation operator we can compute the first few contact terms. We find
\beqa
\mathbb{C}_{\bullet\bullet}^{(2)}(p_1,p_2)\!\!\!&=&\! \!\! \text{see equation (\ref{C2})} \nonumber \\
\mathbb{C}_{\bullet\bullet\bullet}^{(2)}(p_1,p_2,p_3)\!\!\!&=&\! \!\!\frac{2 (\cos (p_1+p_2)+\cos (p_1+p_3)+\cos (p_2+p_3)-2 \cos (p_1+p_2+p_3)-1)}{1-\cos (p_1)-\cos (p_2)-\cos
   (p_3)-\cos (p_1+p_2+p_3)} \nonumber \\
\mathbb{C}_{\bullet\bullet\bullet\bullet}^{(2)}(p_1,p_2,p_3,p_4)\!\!\!&=&\!\!\! \text{see appendix \ref{C4}} \la{contactTerms}
\eeqa
The complexity seems to grow in a scary and uncontrollable fashion. As we will derive later through indirect means, all the contact terms can be written very explicitly as ratios of complicated sums, see appendix \ref{codeContact}. In practice for $N>4$ it takes forever to compute these sums analytically even using a computer (numerically one can easily go up to much larger $N$'s).
For now, let us continue our experimental investigation and turn our attention to more physical quantities such as the structure constants,
in the hope that the huge mess in the contact terms will somehow miraculously drop out. As argued in the introduction this should be the case and indeed it will be.

Finally, let us end by recalling that the momenta of these Bethe eigenstates is quantized through the Bethe equations (\ref{BetheEq0}) which follows from the periodicity of the many particle wave function. The energy of the states is given by (\ref{gamma0}).
\section{Coordinate Bethe Ansatz. Educated Guesses. } \la{coorSec}

To compute one loop structure constants we need control over the spin chain scalar product and expectation values such as the ones in (\ref{tocompute}). We know them at tree level, that is for the one loop eigenvectors of the Dilatation operator. The purpose of this section is to guess their higher loop generalization by doing experiments. In particular, these experiments will allow us to
find an analytic expression for the simplest nontrivial structure constant (corresponding to the case when ${\mathcal O}_1$ is BPS).

We denote the momenta of the Bethe roots of ${\mathcal O}_1$, ${\mathcal O}_2$ and ${\mathcal O}_3$ as ${\bf k}$, ${\bf p }$ and ${\bf q}$ respectively.  Sometimes we shall use $|{\bf 2}\>$ and sometimes we will instead write $|{\bf p}\>$ etc.

\subsection{All Loop Norm Conjecture} \la{normsec}
The simplest scalar products appearing in our list (\ref{tocompute}) are the norms of the Bethe eigenstates.\footnote{Usually a norm is  not a physical object; it depends on the normalization of the state.
In the case of the Bethe sates, however, the normalization is fixed uniquely by the clear physical picture of scattering of the magnons.
Namely, we normalize the state by fixing to $1$ the coefficient of the incident wave, see e.g. (\ref{2magnon}).
After this the norm is well defined and is entitled with some physical meaning which is to be uncovered.}
At one loop the norm of a Bethe eigenstate is given by Gaudin's determinant \cite{Gaudin,paper1}
\begin{equation}
\<{\bf p}| {\bf p}\>_\text{1 loop}=\[\prod_{i<j} \frac{S^{(0)}(p_i,p_j)}{S^{(0)}(p_i^*,p_j^*)}\]^{1/2} \det_{1\le j,k \le N} \[\frac{\partial}{\partial p_j} \( L p_k + \sum_{l \neq k} \frac{1}{i} \log S^{(0)}(p_l,p_k) \)\]  + \mathcal{O}(\lambda) \la{Norm0}
\end{equation}
where ${\bf p }=\{p_j\}$ are the momenta of the $N$ magnons and $L$ is the total length of the operator. At one loop the S-matrix is given by the leading order in (\ref{Smatrix}) which we denoted by $S^{(0)}$. Finally $p_i^*$ is the
complex conjugate of $p_i$; the first factor in (\ref{Norm0}) is absent for real momenta.  Note that the expression inside the determinant is nothing but the left-hand side of Bethe equations (\ref{BetheEq0}) at leading one loop order.
A natural guess for the higher loop generalization is that we simply have to correct the S-matrix,
\begin{equation}
\<{\bf p}| {\bf p}\>_\text{all loops}\stackrel{?}{=} \[\prod_{i<j} \frac{S(p_i,p_j)}{S(p_i^*,p_j^*)}\]^{1/2}  \det_{1\le j,k \le N} \[\frac{\partial}{\partial p_j} \( L p_k + \sum_{l \neq k} \frac{1}{i} \log S(p_l,p_k) \)\]   \la{Norm}\;.
\end{equation}
At  first  this guess may look oversimplified.
After all, when going to higher loops we also needed to include the contact terms (\ref{contactTerms}) which were quite complicated
and one could suspect that the above result is correct modulo these contact terms.
A simple way to settle this question is to perform some {experiments}.\footnote{Latter we will derive (\ref{Norm}) for two loops from an algebraic Bethe ansatz approach where no guesswork will be involved.}
We computed the norm for several different eigenstates with different number of magnons by brute force up to $4$ loops.
Remarkably, we find that (\ref{Norm}) works perfectly when the contact terms are included!
On the contrary, it is \textit{because of the very complicated contact terms} that the final result turns out to be so remarkably simple!
This clearly indicates that some deeper structure must be hidden; unveiling it (at least at two loops) will be the purpose of later sections.
Remarkably, at 4 loops the dressing phase is nontrivial and affects the eigenstate.
Nevertheless we find that (\ref{Norm}) works perfectly!
This leads to the
conjecture:

\textit{For large enough operators, in the usual asymptotic sense, (\ref{Norm}) holds to all loops. }

\subsection{Double Vacuum Decay Amplitudes and Structure Constants}
We now have control over the \verb"norms" contribution in (\ref{C123}), see previous section. We will now move to the remaining contributions: \verb"simple" and \verb"involved". In this section we will
\begin{itemize}
\item Compute \verb"simple"
\item Compute \verb"involved" \textit{for} the simpler cases where ${\mathcal O}_1$ is a BPS operator.
\end{itemize}
It turns out that for ${\mathcal O}_1$ BPS, the contribution \verb"involved" is as simple as the \verb"simple" term. More precisely, these quantities turn out to be related to an important scalar product which we call the Double Vacuum Decay (DVD) Amplitude, see below. Guessing the higher loop expression for this scalar product is the main goal of the current section.

\subsubsection{The Double Vacuum Decay (DVD) Amplitude} \la{simplificationSec}
We will now introduce an important building block of the structure constants which we call Double Vacuum Decay (DVD) amplitude.

To define it let us describe zero energy states of the dilatation operator. We will work with a particular basis of zero energy states which are the so called vacuum descendents of length $L$ and $N$ spin flips.
We denote them as
\begin{equation}
\left|\,\ssststile{\,\,\,\,L\,\,\,\,}{\,\,N\,\,}\,\right\> \equiv
|\! \underbrace{\uparrow\dots\uparrow}_{L-N} \underbrace{\downarrow\dots\downarrow}_{N}\,\>+\text{all $\binom{L}{N}$ permutations}\;.
\end{equation}
We can glue two such vacuum states one after the other making what we call a Double Vacuum $\left|\,\ssststile{\,\,\,\,L\,\,\,\,}{\,\,N\,\,}\,\,\ssststile{\,\,\,\,L'\,\,\,\,}{\,\,N'\,\,}\,\right\>\equiv \left|\,\ssststile{\,\,\,\,L\,\,\,\,}{\,\,N\,\,}\,\right\> \otimes \left|\,\ssststile{\,\,\,\,L'\,\,\,\,}{\,\,N'\,\,}\,\right\>$.
The DVD amplitude is defined as a scalar product between such double vacuum and a Bethe state
\begin{equation}
\mathcal{A}_{L}({\bf q})\equiv (-1)^N\left\<\,\ssststile{\,\,\,\,L\,\,\,\,}{\,\,N'-N\,\,}\,\ssststile{\,\,\,\,L'-L\,\,\,\,}{\,\,N\,\,}\Big| \,{\bf q}\right\> \la{Adef}
\end{equation}
Here $L'$ is the length of the Bethe eigenstate $|{\bf q}\>$ and $N'$ is the number of magnons in it.
If all momenta $\bf q$ are nonzero the Bethe eigenstate is a highest weight state. In this case the DVD amplitude does not depend on $N$.\footnote{
The Bethe state is annihilated by $S^+$. At the same time the adjoint operator $S^-$
acting on the bra state would mix states with two various $N$ from which we conclude that they must be equal up to a sign.}

The DVD amplitude (\ref{Adef}) is an important building block for the structure constants as we will now illustrate.
For example,
the \verb"simple" contribution in (\ref{tocompute})
in the new notation becomes
\begin{equation}
\verb"simple"=\left\<\,\ssststile{\,\,\,\,N_3\,\,\,\,}{\,\,N_3\,\,}\,\ssststile{\,\,\,\,L_3-N_3\,\,\,\,}{\,\,0\,\,}\,\right|1-
g^2 \( \mathbb{H}_{N_{3},N_{3}+1}+ \mathbb{H}_{L_3,1}\) \Big|{\bf 3}\Big\>\la{tocompute2} \,.
\end{equation}
Note that the bra state is annihilated by  $\mathbb{H}_{i,i+1}$ unless $(i,i+1)=(N_3,N_3+1)$ or $(L_3,1)$. This means that we can replace $ \mathbb{H}_{L_{13},L_{13}+1}+ \mathbb{H}_{L_3,1}$ by $\sum_{i=1}^{L_3} \mathbb{H}_{i,i+1}$ in (\ref{tocompute2}). In this way we recognize the one loop dilation operator, see (\ref{dilatation0}), up to a factor of $1/2$.  Then we act with this operator to the right, on $|{\bf 3}\>$. At this loop order, we simply get (one half of) the energy of the Bethe eigenstate
\begin{equation}
\gamma({\bf q})= \sum_{j=1}^{N_3} 8 g^2 \sin^2\frac{q_j}{2} + \mathcal{O}(g^4) \la{Energy}
\end{equation}
where ${\bf q}=\{q_j\}$ are the momenta of the magnons of the operator $\mathcal{O}_3$.
We conclude that
\begin{equation}
\verb"simple"= \(1-\frac{1}{2} \gamma({\bf q})\) \mathcal{A}_{N_3}({\bf q})\;. \la{simplefinal}
\end{equation}

Physically, $\mathcal{A}_L$ describes the probability amplitude for all the $N$ magnons in $|{\bf q}\>$ to occupy homogeneously the first $L$ sites of the chain which has total length $L'>L$. Above $N=N_3$ and $L=N_3$ as well which means so that magnons are all squeezed together in the first $N_3$ sites.  The scalar product $\mathcal{A}_{L}({\bf q})$ was studied in detail in \cite{paper3} and will be considered again in section \ref{Asec}.

\subsubsection{Three Point Functions from DVD Amplitudes} \la{decaysec}
In the previous sections we computed the first contribution \verb"norms" in (\ref{C123}) and reduced the second contribution, \verb"simple", to the computation of the DVD amplitude $\mathcal{A}_{L}({\bf q})$ defined in (\ref{Adef}). We now turn our attention to the \verb"involved" contribution in (\ref{tocompute}).

In general, it is not possible to get rid of the Hamiltonian insertions in this quantity as easily as we did for the contribution \verb"simple". However, if $\mathcal{O}_1$ is a BPS state then we can still do it. In this case, we can follow the steps of the previous section to obtain
\begin{equation}
\verb"involved"=\(1-\frac{1}{2}\gamma({\bf p})\) \mathcal{A}_{L_1-N_3}({\bf p}) \la{hereitis}
\end{equation}
where ${\bf p}=\{p_1,\dots,p_{N_2}\}$ are the momenta of the magnons of the operator $\mathcal{O}_2$. For details see Appendix \ref{422ap}. Combining this result with (\ref{simplefinal}) we conclude that for $\mathcal{O}_1$ BPS we have
\beqa
\nonumber\text{${\mathcal O}_1$ BPS:}\qquad C_{123}^{\circ\bullet\bullet}({\bf p},{\bf q})&=& {\sqrt{L_1L_2L_3}}{}
\(1-\frac{\gamma({\bf p})+\gamma({\bf q})}{2}\) \frac{ \mathcal{A}_{N_3}({\bf q})  \mathcal{A}_{L_1-N_3}({\bf p}) }{\sqrt{\binom{L_1}{N_1} \<{\bf 2}|{\bf 2}\>\<{\bf 3}|{\bf 3}\>}}  \la{C123oxx}\\
\text{ ${\mathcal O}_1$ and ${\mathcal O}_3$ BPS:}\qquad C_{123}^{\circ\bullet\circ}({\bf p},{\bf q})&=& {\sqrt{L_1L_2L_3}}
\(1-\frac{\gamma({\bf p})}{2}\) \frac{ \mathcal{A}_{L_1-N_3}({\bf p}) }{\sqrt{\binom{L_1}{N_1}\binom{L_3}{N_3}\<{\bf 2}|{\bf 2}\>}}  \la{C123oxo}\;.
\eeqa
We already found the norms in section (\ref{normsec}). We now turn to the DVD amplitude $\mathcal{A}$.
\subsubsection{Conjectures for the DVD Amplitude} \la{Asec}
In this section we find an explicit expression for the DVD Amplitude
defined in (\ref{Adef}) valid at tree level and at one loop.
It is the last missing piece to obtain the full one loop corrected structure constants of the type (\ref{C123oxx}).
 At tree level -- i.e. for one loop Bethe eigenstates --we have \cite{paper1}\footnote{As shown by Kostov, the sum (\ref{Asum}) can be recast as a determinant \cite{Ivan},
$$\sum\limits_{\alpha \cup \bar\alpha= \{p\}} (-1)^{|\alpha|} \prod\limits_{k \in \alpha, \bar k \in \bar \alpha} \! \! f(k,\bar k) \,\prod\limits_{\bar k \in \bar\alpha} e^{i L {\bar k}}=\det_{1\le j,k \le N} \[u(p_j)^{k-1}-e^{i p_j L} (u(p_j)-i)^{k-1}\]/\det_{1\le j,k \le N} \[u(p_j)^{k-1}\]\,.$$
 }
\begin{equation}
\mathcal{A}_{\text{tree level}}({\bf p})= \frac{\displaystyle\sum_{\alpha \cup \bar\alpha= \{p\}} (-1)^{|\alpha|} \! \! \prod_{k \in \alpha, \bar k \in \bar \alpha} \! \! f(k,\bar k) \,\prod_{\bar k \in \bar\alpha} e^{i L {\bar k}} }{\displaystyle\prod\limits_{j=1}^N \(e^{-ip_j}-1\) \prod\limits_{1\le i<j \le N} f(p_i,p_j)} \,. \la{Asum}
\end{equation}
The sum is over all possible partitions of the set of momenta and $|\bar\alpha|$ stands for the number of elements in the partition $\bar\alpha$.
For example, for two particles, we have
\begin{equation}
\mathcal{A}_{\text{tree level}}(p_1,p_2)= \frac{1 - f(p_1,p_2) e^{i L p_2}-f(p_2,p_1) e^{i L p_1}+e^{i L (p_1+p_2)} }{(e^{-ip_1}-1)(e^{-ip_2}-1)f(p_1,p_2)}  \la{A2}\;.
\end{equation}
Finally, $\displaystyle f(p,k)=1+i \[u(p)-u(k) \]^{-1}$
where $u(p)$ was defined in (\ref{Smatrix}). We will repeat the game of section \ref{normsec} and try to guess what the higher loop generalization of (\ref{Asum}) could be. The first natural guess is that this expectation value still takes the form (\ref{Asum}) where we simply need to correct $f(p,k)$. To check this proposal we computed (\ref{Adef}) for a two-loop corrected operator $\mathcal{O}_2$ with two magnons. For that we use the two loop wave functions, with contact terms, as described in the previous section, see appendix \ref{ApA1} for details. We realize that (\ref{Adef}) indeed still takes the form (\ref{A2}) with a corrected $f(p,k) \to \mathfrak f(p,k)$,
\begin{equation}
\mathfrak{f}(p,k)=\frac{u(p)-u(k)+i}{u(p)-u(k)} \(1+\frac{g^2}{(u(p)^2+\frac{1}{4})(u(k)^2+\frac{1}{4})} + \mathcal{O}(g^4)\) \,. \la{fcorr}
\end{equation}
We should emphasize that this is far from obvious. In fact, it is because of the particular form of the contact terms that the result can still be put in the form (\ref{A2}). By computing (\ref{Adef}) for a few magnons we verify that
\begin{equation}
\mathcal{A}_L({\bf p})= \frac{\displaystyle\sum_{\alpha \cup \bar\alpha= \{p\}} (-1)^{|\alpha|} \! \! \prod_{k \in \alpha, \bar k \in \bar \alpha} \! \! \mathfrak{f}(k,\bar k) \,\prod_{\bar k \in \bar\alpha} e^{i L {\bar k}} }{\displaystyle\prod\limits_{j=1}^N \(e^{-ip_j}-1\) \prod\limits_{1\le i<j \le N} \mathfrak{f}(p_i,p_j)}  \la{Afin}
\end{equation}
works perfectly at one loop! In fact, by appropriately correcting (\ref{fcorr}) we seem to be able to reproduce the two and three loop results \cite{GSV}. This leads to the next general conjecture:

\textit{For large enough operators, in the usual asymptotic sense, (\ref{Afin}) holds to all loops provided (\ref{fcorr}) is properly fixed. }\footnote{
At higher loops the Dilatation operator has scheme dependent terms which cannot be fixed by  symmetry and locality, see \cite{Beisert:2007hz}.
These terms have unfixed coefficients and do not affect the spectrum. They can be removed by a similarity transformation.
The DVD amplitude is not invariant under such transformations and can, in principle, contain some scheme dependence which of course should cancel out in physical quantities.
This subtlety will be considered elsewhere.}

For more details on how to fix $\mathfrak{f}(p,k)$ see appendix \ref{ApA2}. In particular, in this appendix, we present an alternative procedure which
constrains this function to all loop orders.

This concludes our study of the simplest correlation functions in (\ref{C123oxo}) where ${\mathcal O}_1$ is BPS. We conjectured the form of all the building blocks appearing in this expression and even proposed some all loop conjectures.

On the other hand we should conclude this section by mentioning that we failed in guessing most general
structure constants when both ${\cal O}_1$ and ${\cal O}_2$ are non-BPS operators.
This is partly because the tree level result simplifies enormously on-shell i.e. when
the set of roots ${\bf p}$ is constrained by the Bethe ansatz equations.
At the same time this complicates enormously the guesswork since this constraint
is rather complicated to deal with when doing experiments. In sum, the guessing strategy used so far seems insufficient for these more complicated cases.
We need some more scientific and systematic approach to find the one loop (and higher loops) deformation.
Developing such an approach is the main goal of the current paper and will be the focus of the remaining sections. The more impatient readers can check Appendix \ref{fullAfter} for the final  result for the most general one loop structure constants for three non-BPS operators.

\section{Algebraic Bethe Ansatz Approach and Impurities} \la{algSec}

In the previous section we followed the coordinate Bethe ansatz approach where the states are written as sums of plane waves with relative coefficients given by $S$-matrix elements. Beyond one loop we also have contact terms, see section \ref{contactSec}. In this section we will only consider the one loop states for which there is a well developed alternative approach known as the Algebraic Bethe ansatz. In this approach multi-particle states are simply given by
\begin{equation}
\includegraphics[trim=0cm 6cm 0cm 0cm, clip=true, scale=.6]{paths.pdf} \la{stateAlg}
\end{equation}
In fact, the states represented here already contain a new twist: the impurities $\theta_j$ at each of the spin chain sites. To get the one loop Bethe eigenstates we set all impurities to zero, $\theta_j=0$ and consider $|0;{\bf u}\>$. However, as we will see in the next sections, these impurities will play a central role when going to higher loops.

The relation between the Bethe roots $u_j$ and the momenta $p_j$ is given by $u_j=u(p_j)$ where the change of variables was introduced in (\ref{Smatrix}).
This figure is a graphical implementation of the algebraic relation\footnote{More explicitly,
\begin{equation}
\hat B(u) = \left\<\uparrow\right| \bigotimes_{j=1}^L \(\mathbb{I}_{j0} +\frac{i}{u-\theta_j-i/2} \mathbb{P}_{j0} \)\left|\downarrow\right\>
\end{equation}
Here the subscript $j0$ indicates that the operator acts on the tensor product of physical Hilbert space $V_j = \mathbb{C}^2$ corresponding to spin chain site $j$ and an auxiliary space $V_0=\mathbb{C}^2$. The bra and ket in this formula live in this auxiliary space so that $\hat B(u)$ is an operator acting on the full spin chain Hilbert space $\mathcal{H}=V_1\otimes \dots \otimes V_L$. Graphically, the action of many such operators can be represented as in (\ref{stateAlg}). For algebraic Bethe ansatz reviews see for example \cite{paper1} and \cite{omar}.}
\begin{equation}
|\theta;{\bf u}\>=\hat B(u_1) \dots \hat B(u_N) \left|\uparrow\dots\uparrow\right\> \la{BBOmega}\;.
\end{equation}
It is quite remarkable that such creation operators $\hat B(u_j)$ exist at all. Even though  the different $\hat B$'s enter independently
they create fully interacting particles!
For example, for the two particle state, we obtain the following wave function (we have already set $\theta_j=0$ in this figure):
\begin{equation}
\includegraphics[trim=0cm 13cm 0cm 0cm, clip=true, scale=.6]{2magnons.pdf} \la{2ptExample}
\end{equation}
which leads, up to a simple proportionality factor, to  the exactly the same wave function $\psi_{coor}(n_1,n_2)$ found through the Coordinate Bethe Ansatz approach,
\begin{equation}
\psi_{coor}(n_1,n_2)= e^{ip_1n_1+ip_2n_2}+S(p_2,p_1) e^{ip_2n_1+ip_1 n_2} \,,
\end{equation}
see (\ref{2magnon}) with $g^2=0$. Of course, this continues to hold for more magnons. For $N$ magnons, the relative normalization between the one loop coordinate states $|{\bf p}\>_\text{one loop}$ and algebraic states $|0;{\bf u}\>$ is given by
\begin{equation}
|0;{\bf u}\>=\mu\, |{\bf p}\>_\text{one loop} \,, \qquad \mu\equiv\prod_{j=1}^N (e^{-ip_j}-1) \prod_{j<k}^N f(p_j,p_k) \,. \la{lambda}
\end{equation}
where $f(p,k)=1+i/[u(p)-u(k)]$.
Obviously, at the end of the day, all normalizations drop out for physical quantities.\footnote{As we emphasized
above the coordinate Bethe ansatz normalization is particular nice from a physical point of view
and we would like to be able to derive at least the first several loop orders for the norm. In (\ref{Norm}) we conjectured a value for the norm at all-loops in the coordinate Bethe ansatz normalization.}

\subsection{Impure States and Structure Constants} \la{impure}
To go to higher loops we need some sort of deformation of (\ref{BBOmega}) consistent with Integrability. The simplest one, which basically costs us nothing, is to introduce so called impurities $\theta_j$ at each lattice site $j$. In fact, in the previous section we already introduced these parameters, see (\ref{stateAlg}). At one loop we simply set those impurities to zero. The idea -- which will be explored in detail in the following sections -- is to use these impurities to mimic the loop corrected propagation and interaction of the magnon excitations. This deformation will amount to acting on the states (\ref{stateAlg}) with a differential operator made out of $\partial/\partial \theta_j$ and only then setting all impurities to zero at the end.

Concerning the structure constants, this procedure will allow us to promote tree level results with impurities to higher loop results without impurities. For future use let us quote here what the three point functions at tree level look like once we introduce the impurities, see figure \ref{figC123boxes}. This was worked out in \cite{omar}, see also \cite{short}.

Let us make a simplifying assumption that $\mathcal{O}_3$ is BPS. In any event, the contribution of $\mathcal{O}_3$ factorizes into a \verb"simple" contribution -- see section \ref{setup} -- so it is straightforward to generalize to the case with ${\mathcal O}_3$ non-BPS at any later stage if needed.

We use $u_j=u(p_j)$ and $v_j=u(k_j)$ for the rapidities of operators  $\mathcal{O}_1$ and ${\mathcal O}_2$ respectively.

Finally we have to introduce some notation for the impurities. We denote the impurities of each operator ${\mathcal O}_a$ by $\theta_j^{(a)}$. Each impurity is associated to a line going from one operator to another so it is shared between two operators. We denote the set of impurities shared between ${\mathcal O}_a$ and ${\mathcal O}_b$ by $\theta^{(ab)}$ so that, for example, $\theta^{(2)}=\theta^{(12)}  \cup\theta^{(23)} $ and so on.
When omitting the superscript it is implicit that we refer to the first operator, $\theta_j\equiv \theta_j^{(1)}$.

Then we have
\begin{figure}[t]
\begin{center}
\includegraphics[trim=9cm 1cm 0cm 0cm, clip=true, scale=.5]{C123paths3.pdf}
\end{center}
\caption{Tree level impure structure constant. Each box represents a sum of paths as in (\ref{stateAlg}). We should also divide by the normalization of the three operators/spin chains. } \la{figC123boxes}
\end{figure}
\begin{equation}
|{C}^{\rm tree\;with\;imp.}_{123}| = \frac{\sqrt{L_1L_2L_3}}{\sqrt{\binom{L_3}{N_3}}}  \, \frac{|\<\theta^{(1)};{\bf u} | \hat{\mathcal{O}}_3| \theta^{(2)};{\bf v} \>|}{\sqrt{\left\<\theta^{(1)};{\bf u} |\theta^{(1)};{\bf u}\right\>\left\<\theta^{(2)};{\bf v} | \theta^{(2)};{\bf v} \right\>}}  \label{main} \,.
\end{equation}
where
\begin{equation}
\left\<\theta;{\bf u} |\theta;{\bf u}\right\>=\prod_{m\neq k} \frac{u_k-u_m+i}{u_k-u_m}   \det_{j,k\le N_1} \frac{\partial }{\partial u_j} \[ \frac{1}{i} \sum_{a=1}^{L_1} \log \frac{u_k-\theta_a+i/2}{u_k-\theta_a-i/2}
+\frac{1}{i}\sum_{m\neq k}^{N_1} \log\frac{u_k-u_m-i}{u_k-u_m+i}\]  \la{normB}
\end{equation}
with a similar expression for the other norm. Before presenting the numerator in (\ref{main}) let us discuss (\ref{normB}) briefly and make contact with the previous sections. The expression (\ref{normB}) is the norm of Bethe eigenstates with impurities. The Bethe equations with impurities take the form
\begin{equation}
 \sum_{a=1}^{L_1} \frac{1}{i} \log\(\frac{u_k-\theta_a+i/2}{u_k-\theta_a-i/2}\)+
\sum_{m\neq k}^{N_1} \frac{1}{i}\log\( \frac{u_k-u_m-i}{u_k-u_m+i}\) = 2\pi n_k \,, \qquad n_k \in \mathbb{Z} \la{BAEimpure}
\end{equation}
which reduces to (\ref{BetheEq0}) if we set all impurities to zero. Hence the determinant in (\ref{normB}) resembles very much (\ref{Norm0}). There is however a different prefactor which arises because (\ref{normB}) was computed using the Algebraic Bethe Ansatz normalization whereas (\ref{Norm0}) was obtained in the coordinate Bethe Ansatz formalism. Of course, in (\ref{main}), the normalizations of the kets are irrelevant. However, to compare the intermediate quantities (\ref{normB}) and (\ref{Norm0}) we better know the relation between the two. When there are no impurities the relative normalization is (\ref{lambda}). Taking it into account we see that (\ref{normB}) and (\ref{Norm0}) agree in the homogeneous limit $\theta_j\to 0$.
\begin{figure}[t]
\begin{center}
\includegraphics[trim=0cm 9cm 0cm 0cm, clip=true, scale=.6]{figimp.pdf}
\end{center}
\caption{Main scalar product appearing in the impure structure constants (\ref{omar}). When contracting the three operators the rightmost sites of the spin chain ${\mathcal O}_2$ are contracted with empty sites in the chain of the third operator. Hence the weight associated with the lower right corner in the right picture is just $1$ since the weight for each vertex where a blue thick lines crosses a thin gray line is $1$, see (\ref{stateAlg}). Because of this, this scalar product does not depend on the impurities $\theta^{(23)}$ at all and can be reduced to the object in the right \cite{omar}.
} \la{figC123}
\end{figure}

Finally we have the numerator which can be represented graphically as in figure \ref{figC123}. We have \cite{omar}
\begin{equation}
\<\theta^{(1)} ;{\bf u} | \hat{\mathcal{O}}_3| \theta^{(2)};{\bf v} \>=\frac{\displaystyle
\prod_{m}^{N_3}\prod_{n}^{N_1}(u_n- \hat\theta^{(1)}_{m}+i/2)/\prod_{m}^{N_3}\prod_{n}^{N_2}(v_n- \hat\theta^{(1)}_{m}+i/2)
}{\displaystyle
\prod_{n<m}^{N_1}(u_m-u_n)
\prod_{n<m}^{N_2}(v_n-v_m)
\prod_{n<m}^{N_3}( \hat\theta^{(1)}_{n}-\hat\theta^{(1)}_{m}) }
\det\Big([G_{nm}] \oplus [F_{nm}] \Big) \la{omar}
\end{equation} where $\hat \theta^{(1)}_m=\theta^{(1)}_{L_1+1-m}$ and
\begin{equation}
F_{nm}=\frac{1}{(u_{n}-{\color{black}\hat\theta}_m^{(1)})^2+\frac{1}{4}} \,, \qquad G_{nm}=
\prod_{a=1}^{L_1} \frac{v_m-\theta_a^{(1)}+i/2}{v_m-\theta_a^{(1)}-i/2}
\frac{\prod_{k\neq{   n}}^{N_1}(u_k-v_{   m}\!+\! i)}{u_n-v_m}-
\frac{\prod_{k\neq{   n}}^{N_1}(u_k-v_{   m}\!-\! i)}{u_{   n}-v_{   m}} \nonumber\;.
\end{equation}
Where $F_{nm}$ is $N_1\times N_3$ and $G_{nm}$ is $N_1\times N_2$ matrices
completing together to a $N_1\times N_1$ square matrix.  Note that only the $\theta^{(1)}=\theta^{(12)}\cup\theta^{(13)}$ appears in the numerator factor. There is no dependence on the impurities $\theta^{(23)}$ associated to lines going from ${\mathcal O}_2$ to ${\mathcal O}_3$. From the combinatorial point of view this is obvious, see caption of figure \ref{figC123}.

The homogeneous limit is a bit more tricky here, see \cite{omar} for details.
One has to resolve a $0/0$ uncertainty using standard l'H\^opital's rule.
On the one hand the prefactor in (\ref{omar}) diverges as $\theta_j\to 0$. On the other hand the determinant vanishes since $F_{nm}$ will no longer depend on the column index $m$ as $\theta_m\to 0$.\footnote{Once we take the limit carefully the result is proportional to  the determinant
\beqa
\left|\begin{array}{cccccc}
\frac{\partial T(v_1)}{\partial u_1} & \dots &  \frac{\partial T(v_{N_2})}{\partial u_1} & q_2(u_1) & \dots & q_{N_3+1}(u_1)\\
\vdots & & \vdots & \vdots & & \vdots \\
\frac{\partial T(v_1)}{\partial u_{N_1}} & \dots &  \frac{\partial T(v_{N_2})}{\partial u_{N_1}} & q_2(u_{N_1}) & \dots & q_{N_3+1}(u_{N_1})
\end{array}
\right| \,\,, \qquad \begin{array}{l}
q_n(u)\equiv i(u+i/2)^{-n+1}-i(u+i/2)^{-n+1}\\
T(u)\equiv \frac{Q(u-i)}{Q(u)} + \(\frac{u-i/2}{u+i/2}\)^{L_1} \frac{Q(u+i)}{Q(u)}\\
Q(u) \equiv \prod_{j=1}^{N_1} (u-u_j)
\end{array}
\la{theDet}
\eeqa
The expression (\ref{theDet}) is the most complicated factor in the tree-level structure constant for three non-BPS operators. In section (\ref{coorSec}) we tried to guess the higher loop generalization of several simpler scalar products rather successfully. Guessing the generalization of (\ref{theDet}) turns out to be quite non-trivial as mentioned before.
}

In what follows we will also need a particular case of the general scalar product (\ref{main}).
That is the case when ${\cal O}_1$ degenerates to BPS.
This can be done by sending all roots $u_j \to \infty$. In this limit
 the state $|1\>$ becomes a vacuum descendent and (\ref{omar}) reduces to the generalization of the DVD amplitude with impurities,
\begin{equation}
\left\<\,\ssststile{\,\,\,\,L_1-N_3\,\,\,\,}{\,\,N_1-N_3\,\,}\,\,\ssststile{\,\,\,\,L_3-N_3\,\,\,\,}{\,\,0\,\,}\,\right|
 \theta^{(2)};{\bf v}\Big \>=\sum_{\alpha \cup \bar \alpha = {\bf v}} (-1)^{|\bar \alpha|} \[\prod_{\bar a \in \bar\alpha}  \prod_{n=1}^{L_{12}} \frac{\bar a-\theta_n+i/2}{\bar a-\theta_n-i/2}\] \prod_{a\in \alpha\,, \bar a\in \bar \alpha} f(a,\bar a)
\end{equation}
where $f(u,v)=1+i/(u-v)$. Denoting the square bracket by $e_{\bar \alpha}^{L_{12}}$ and using the same kind of short-hand notation from \cite{paper1} we can compress this expression to
\begin{equation}\la{tree132}
\<\text{vac}|
 \theta^{(2)};{\bf v}\> =
\sum_{\alpha \cup \bar \alpha = {\bf v}}
(-1)^{|\bar\alpha|}
f^{\alpha,\bar\alpha}e_{\bar \alpha}^{L_{12}}\,,\,\,\, \text{where} \qquad \<\text{vac}|=\left\<\,\ssststile{\,\,\,\,L_1-N_3\,\,\,\,}{\,\,N_1-N_3\,\,}\,\,\ssststile{\,\,\,\,L_3-N_3\,\,\,\,}{\,\,0\,\,}\,\right|\;.
\end{equation}
Equation (\ref{main}) -- and also equations (\ref{normB}) and (\ref{tree132}) -- will be used in the next section.
As we will see later, by use of well designed differential operators we will be able to promote  tree level
``impure" expressions to higher loops ``pure" results. Developing the formalism to do so will be the subject of the next section.

\section{$\Theta$--morphism and Quantum Corrected States} \la{secMor1}
In this section we introduce the main object of this paper: the $\Theta$--morphism.
\subsection{One Loop $\Theta$--Morphim. Definition and Properties}

The $\Theta$--morphism is a coupling dependent functional acting on functions of a set of variables $\theta\equiv \{\theta_i\}_{i=1}^L$
which we denote by double brackets
\begin{equation}
\Theta\;:\;f(\theta)\;\mapsto\;\ckappa{\theta}{}{f(\theta)}\;.
\end{equation}

In section \ref{Axiomatics} we will explain how one can define this operator by requiring it to obey several defining axioms.
The justification of the name will be clear from what follows. At leading order the $\Theta$--morphim is given by the differential operator
\begin{equation}
\ckappa{\theta}{}{f(\theta)}\equiv  \left. f+\frac{ g^2}{2}\sum_{i=1}^{L} \mathcal{D}_i^2 f + \mathcal{O}(g^4) \right|_{\theta_j \to 0}\;. \la{mor1}
\end{equation}
where
\begin{equation}
\mathcal{D}_i = \partial_i- \partial_{i+1} \qquad \text{and} \qquad  \partial_i \equiv \frac{\partial}{\partial \theta_i}\,.
\end{equation}
we use a periodic boundary conditions so that $\mathcal{D}_L=\partial_L-\partial_1$.
For now let us take the definition (\ref{mor1}) as our starting point. For further motivation see the next subsections, \cite{short} and \cite{GSV}.
In particular, we will see below that by acting with this operator on the impure states of section \ref{impure}, we generate perfect two loop eigenvectors of the Dilatation operator (\ref{dilatation0}) including all the messy contact terms discussed in section \ref{contactSec} and appendix \ref{C4}! This is of course the main justification for (\ref{mor1}).

For now, let us explore some properties of the map (\ref{mor1}) that will come in handy later on. Acting on a product of two functions we have
\begin{equation}
\ckappa{\theta}{}{f(\theta)h(\theta)}=
\ckappa{\theta}{}{f(\theta)}
\ckappa{\theta}{}{h(\theta)}+
\cckappa{\theta}{}{f(\theta)}{h(\theta)}
\;. \la{mor0}
\end{equation}
where
\begin{equation}
\cckappa{\theta}{}{f}{h}\equiv g^2\sum_{i=1}^{L} \mathcal{D}_i f \,\mathcal{D}_i h\,. \la{ct}
\end{equation}
In the right hand side we  set to zero all impurities after taking the derivatives, just like we did in (\ref{mor1}). In what follows this is always implicit; we always set $\theta_j\to 0$ at the end. We denote the contribution (\ref{ct}) by \textit{cross-term}. If it were not for this term then (\ref{mor0}) would be the mathematical definition of a morphism
\footnote{We are grateful to B.~Vicedo for proposing this name to us}
. Hence, the cross-term is the deviation from the morphism property.

The cross-term (\ref{ct}) does vanish if one of the functions is a totally symmetric function of the $\theta_i$. That is we do have the restricted morphism relation
\begin{equation}
{\bf Morphism:}\;\;\;\;\ckappa{\theta}{}{f_{\rm sym}(\theta)h(\theta)}=
\ckappa{\theta}{}{f_{\rm sym}(\theta)}
\ckappa{\theta}{}{h(\theta)}\;. \la{morpro}
\end{equation}
Another nice property that will be used shortly is that
\begin{equation}
{\bf Dispertion:}\;\;\;\;\ckappa{\theta}{}{\sum_{i=1}^L\log\frac{u-\theta_i+\tfrac i2}{u-\theta_i-\tfrac i2}}=
L\log\frac{x(u+\tfrac i2)}{x(u-\tfrac i2)} + \mathcal{O}(g^4)\;\, , \la{disppro}
\end{equation}
which one can easilly check directly using (\ref{mor1}). In this expression $x(u)$ is the so called Zhukowsky variable
\begin{equation}
x(u)+\frac{1}{x(u)} \equiv \frac{u}{g} \,, \qquad x(u) = \frac{u}{g}-\frac{g}{u}+\mathcal{O}(g^3) \,. \nonumber
\end{equation}
These relations are very important and have  interesting implications.
 For example, an obvious consequence of the morphism relation (\ref{morpro}) is that for any function $F$ we have
\begin{equation}
\ckappa{\theta}{}{F \Big[ f_{\rm sym}(\theta) \Big]}=
F\[\ckappa{\theta}{}{f_{\rm sym}(\theta)}\] \,. \la{lema}
\end{equation}
Another nice implication follows when we combine both properties:  Suppose we are dealing -- as we often will -- with a function $f( { \bf u}, \theta)$ which is also a function of
 the Bethe roots ${\bf u} = \{u_1,\dots, u_N\}$. These are quantized according to Bethe equations (\ref{BAEimpure}),
\begin{equation}
\sum_{a=1}^{L} \frac{1}{i} \log\(\frac{u_k-\theta_a+i/2}{u_k-\theta_a-i/2}\)+
\sum_{m\neq k}^{N} \frac{1}{i}\log\( \frac{u_k-u_m-i}{u_k-u_m+i}\) = 2\pi n_k \,, \qquad n_k \in \mathbb{Z} \la{notanymore}
\end{equation}
In particular, they depend on the impurities $\theta_j$ in a totally symmetric way.
Then, a consequence of the above relations is that $\ckappa{\theta}{}{f( { \bf u}, \theta)}$ is still given by (\ref{mor1}) where
\begin{enumerate}
\item we can treat the $u_j$ as independent of $\theta_j$ when taking the derivatives in (\ref{mor1}) \textit{but}
\item after applying the morphism -- i.e. in the right hand side of (\ref{mor1}) -- the Bethe roots are now quantized according to the BDS equations \cite{BDS}
\begin{equation}
\phi_k^{\bf u} \equiv \frac{L}{i} \log\frac{x(u_k+i/2)}{x(u_k-i/2)}+
\sum_{m\neq k}^{N} \frac{1}{i}\log\( \frac{u_k-u_m-i}{u_k-u_m+i}\) = 2\pi n_k \,, \qquad n_k \in \mathbb{Z} \la{BAEimpure2}
\end{equation}
instead of (\ref{notanymore}). The equations (\ref{BAEimpure2}) are nothing but the corrected Bethe equations (\ref{BetheEq0}) and (\ref{Smatrix}). \end{enumerate}
Of course, such a prescription is exactly what one would like when dealing with expressions involving Bethe roots!

\subsection{Single Derivative of Monodromy Matrix et al} \la{IdSec}
The most important object in the Algebraic Bethe ansatz construction is the monodromy matrix
\begin{equation}
\hat L(\theta;u) \equiv  \(\mathbb{I}_{10} +\frac{i}{u-\theta_1-i/2} \mathbb{P}_{10} \) \otimes \dots \otimes \(\mathbb{I}_{L0} +\frac{i}{u-\theta_L-i/2} \mathbb{P}_{L0} \) \la{Ldef}
\end{equation}
Operators with subscript $ij$ act on $V_i \otimes V_j$ where $V_{1},\dots,V_L$ are the Hilbert spaces at each spin chain site while $V_0$ is an auxiliary space which is also $\mathbb{C}^2$. That is, we can think of $\hat L(u)$ as a $2\times 2$ matrix whose entries are operators acting on the full physical Hilbert space $\mathcal{H}=V_1\otimes \dots \otimes V_L$,
\begin{equation}
\hat L(\theta;u) =\( \begin{array}{cc}
\hat A(u) & \hat B(u) \\
\hat C(u) & \hat D(u) \end{array} \)
\end{equation}
The operators $\hat B(u)$ are creation operators; the Bethe eigenstates are simply given by (\ref{BBOmega}). Graphically, this action can be represented as in (\ref{stateAlg}). A very important relation that we will use extensively is then
\begin{equation}
i{\cal D}_j\hat L=-
[\mathbb{H}_{j,j+1}, \hat L]+\delta_{j,L}\sum_{n=1}^L[ \mathbb{H}_{n,n+1}, \hat L] \,. \la{basic}
\end{equation}
where $\mathbb{H}_{j,j+1}=\mathbb{I}_{j,j+1} - \mathbb{P}_{j,j+1}$. As usual, we have set to zero all impurities after taking the derivatives. Hence, when acting on Bethe states $|\theta;{\bf u}\> = \hat B(u_1) \dots \hat B(u_N) \left| \uparrow \dots \uparrow \right\> $ with a single derivative $\mathcal{D}_j$ we get from (\ref{basic})  that $i{\cal D}_j |\theta;{\bf u}\> = - \mathbb{H}_{j,j+1}|0;{\bf u}\> + \delta_{j,L}\sum_{n=1}^L \mathbb{H}_{n,n+1} |0;{\bf u}\>  $
and since they are Bethe eigenstates the last term can be replaced by the state energy so that
\begin{equation}
i{\cal D}_j |\theta;{\bf u}\> = - \mathbb{H}_{j,j+1} |0;{\bf u}\> + \delta_{j,L} \Gamma_{\bf u} |0;{\bf u}\>  \,. \la{PsiId}
\end{equation}
where
\begin{equation}
\Gamma_{\bf u} = \sum_{j=1}^N \frac{1}{u_j^2+\frac{1}{4}} \,.
\end{equation}
Another very important object is the transfer matrix
\begin{equation}
\hat T(\theta;u) \equiv {\rm Tr}\, \hat L(\theta;u) = \hat A(u) + \hat D(u) \,. \la{Tdef}
\end{equation}
To the order we will be working at -- i.e. as an expansion in small $\theta_j$'s -- this object commutes with the Hamiltonian $\sum_{n=1}^L \mathbb{H}_{n,n+1}$. Hence, from (\ref{basic}) we find
\begin{equation}
i{\cal D}_j\hat T=-
[\mathbb{H}_{j,j+1}, \hat T] \,. \la{Tid}
\end{equation}
The identities (\ref{basic}), (\ref{PsiId}) and (\ref{Tid}) will be extensively used in what follows.

\subsection{Integrability and Two Loop Eigenstates} \la{algStatesSec}
Consider the impure states (\ref{BBOmega}).
They diagonalize the transfer matrix (\ref{Tdef}),\footnote{The eigenvalue reads \begin{equation}
{T}(\theta;v)=a(\theta;v)\frac{Q_{\bf u}(v-i)}{Q_{\bf u}(v)}+\frac{Q_{\bf u}(v+i)}{Q_{\bf u}(v)} \,, \text{ where} \,\,\,\,\,
a(\theta;v) \equiv \prod_{a=1}^L \frac{v-\theta_a+i/2}{v-\theta_a-i/2} \,, \,\,\,\,\, Q_{\bf u} (v) \equiv \prod_{j=1}^N (v-u_j) \la{footeq}
\end{equation}
}
\begin{equation}
{\hat T}(\theta;v)|\theta;{\bf u}\rangle={T}(\theta;v)|\theta;{\bf u}\rangle \la{diag} \,.
\end{equation}
The eigenvalue ${T}(\theta;v)$ is a symmetric function of the $\theta_j$'s.
A most striking fact about (\ref{diag}) is that the states $|\theta;{\bf u}\rangle $ do \textit{not} depend on the spectral parameter $v$ and still they diagonalize the full family of operators ${\hat T}(\theta;v)$ for \textit{any} value of $v$! Integrability, that is exact solvability of the model, lies upon this remarkable fact. In this section we want to understand what happens to such remarkable properties  once we apply the $\Theta$--morphism to relations like (\ref{diag}).

Let $\hat F(v)$ be an arbitrary function of
${\hat T}(\theta;v)$ and $F(v)$ its eigenvalue. Then
\begin{equation}
\hat F(v) |\theta;{\bf u}\rangle=F(v) |\theta;{\bf u}\rangle \,.
\end{equation}
Applying the $\Theta$--morphism we get
\begin{equation}
\ckappa{\theta}{}{\hat F}
\ckappa{\theta}{}{ |\theta;{\bf u}\rangle}+
\cckappa{\theta}{}{\hat F}{ |\theta;{\bf u}\rangle}
=
\ckappa{\theta}{}{F}
\ckappa{\theta}{}{|\theta;{\bf u}\rangle}\la{stepone} \,.
\end{equation}
in the right hand side we get no cross term since $F$ is a symmetric function of $\theta$'s, see (\ref{morpro}).
Consider the cross-term in the left-hand side. Using the relations derived in the previous section we have
\begin{equation}
\cckappa{\theta}{}{\hat F}{ |\theta;{\bf u}\rangle}=-g^2\sum_{j=1}^L [P_{j,j+1},{\hat F}]P_{j,j+1}
|0;{\bf u}\rangle
+g^2[{\mathbb H}_{L,1},{\hat F}]\Gamma_{\bf u}
|0;{\bf u}\rangle \la{step2}
\end{equation}
As usual we set the impurities to zero at the end in the right hand side. Note that at this order -- since both terms are already multiplied by a $g^2$ factor -- we can replace either of the kets $|0;{\bf u}\rangle$ by $\ckappa{\theta}{}{|\theta;{\bf u}\rangle}$ if we want to.
Note also that only the last term in (\ref{step2}) does not have have nice cyclicity properties; this term can be interpreted as a correction to the state.
More precisely, if we define
\begin{equation}\la{closed}
|{\bf u}\rangle\equiv
(1-g^2 \Gamma_u {\mathbb H}_{L,1})
\ckappa{\theta}{}{|\theta;{\bf u}\rangle}
\end{equation}
then  (\ref{stepone}) becomes
\begin{equation}
\hat{\mathcal{F}}(v) |{{\bf u}}\rangle =
{\mathcal{F}}(v) |{{\bf u}}\rangle \la{after}
 \end{equation}
where
\begin{equation}
\hat{\mathcal{F}} \equiv \ckappa{\theta}{}{\hat F}
-g^2\sum_{j=1}^L [P_{j,j+1},{\hat F}]P_{j,j+1}
\qquad \text{and}\qquad \mathcal{F} \equiv
 \ckappa{\theta}{}{F} \la{calF}\;.
\end{equation}
The very exciting property we highlighted above is present again in (\ref{after}). Namely, we have a set of states $| {{\bf u}}\>$ which do not depend on the spectral parameter and still diagonalize a full family of operators $\hat{\mathcal F}(v)$ for arbitrary $v$. In this sense, we have integrability before and after the $\Theta$--morphism.

\subsection{Two-loop Hamiltonian}
We found the first sign of evidence in favor of the $\Theta$--morphism in (\ref{BAEimpure2}). In these Bethe equations we identified the corrected momentum
\begin{equation}
p(u) = \frac{1}{i} \log \frac{x(u+i/2)}{x(u-i/2)}\;.
\end{equation}
Now we should look for the corrected energy
\begin{equation}
\epsilon(u) = \frac{2ig}{x(u+i/2)} -\frac{2ig}{x(u-i/2)}
\end{equation}
More precisely we want to find some operator $\hat F$ such that $\hat{\mathcal{F}}$ defined through (\ref{calF}) leads to the two loop Dilatation operator (\ref{dilatation0}). We will look for this operator by fixing the eigenvalue $\mathcal{F}$ to be equal to the energy
\begin{equation}
\gamma=\sum_{j=1}^L\epsilon(u_j) =
\sum_{j=1}^N \[
\frac{2ig^2}{u_j+\tfrac i2}
+
\frac{2ig^4}{(u_j+\tfrac i2)^3}+c.c. \] + \mathcal{O}(g^6) \,.\la{energyc}
\end{equation}
Using (\ref{footeq}) we can directly check that\footnote{We would still get (\ref{energyc}) without the $\Theta$--morphism $\ckappa{\theta}{}{}$, that is if we simply set the impurities to zero and not take any derivatives. Of course, in that case the rapidities would obey (\ref{BAEimpure}) with $\theta_j=0$, that is they would be solutions to the one loop Bethe equations which is not what we want. If we apply the $\Theta$--morphism we do get  the two loop corrected Bethe equations (\ref{BAEimpure2}) as desired. Indeed, as we can see directly from (\ref{three}) below, we would not get the correct two loop Hamiltonian without the morphism contribution.}
\beqa
\gamma=\left.\ckappa{\theta}{}{2ig^2\d_u\log T+ig^4\d_u^3\log T}\right|_{u=-i/2} \, \la{gammaT} \,.
\eeqa
Hence, following (\ref{calF}), the candidate operator that should correspond to the two loop Dilatation operator is\footnote{At this loop order $g^4 i\d_u^3\log \hat T$ and $g^4 i\d_u^3 \ckappa{\theta}{}{\log \hat T}$ are indistinguishable.}
\begin{equation}
\left.\ckappa{\theta}{}{2g^2i\d_u\log \hat T}+g^4 i\d_u^3\log \hat T
-2 g^4\sum_{j=1}^L [P_{j,j+1},i\d_u\log \hat T]P_{j,j+1}
\right|_{u=-i/2} \la{todo}\;.
\end{equation}
Using the definition (\ref{Tdef}) and (\ref{Ldef}) we find indeed that these three contributions lead to
\beqa
&&\left\{\begin{array}{c}
\ckappa{\theta}{}{2g^2i\d_u\log \hat T} \\
g^4 i\d_u^3\log \hat T \\
-2 g^4\sum_{j=1}^L [P_{j,j+1},i\d_u\log \hat T]P_{j,j+1} \end{array}\right\}_{u=-i/2} = \left\{\begin{array}{c}
2g^2-8g^4 \\
-\,4g^4\\
+\,4g^4 \end{array}\right\} \sum_{i=1}^L (\mathbb{I}-\mathbb{P}_{i,i+1})+ \nonumber\\
&&\qquad+ \left\{\begin{array}{c}
+\,4g^4 \\
+\,2g^4\\
-\,4g^4 \end{array}\right\} \sum_{i=1}^L (\mathbb{I}-\mathbb{P}_{i,i+2})+ \left\{\begin{array}{c}
-\,2g^4 \\
+\,2g^4\\
-\, 0\end{array}\right\}\sum_{i=1}^L \[ {\mathbb P}_{i,i+1} , \[ {\mathbb P}_{i+1,i+2}, {\mathbb P}_{i+2,i+3}\]\]  \la{three}
\eeqa
Adding the three quantities we see that (\ref{todo}) is exactly equal to
\begin{equation}
\mathbb{H}=(2g^2-8g^4)\sum_{i=1}^L (\mathbb{I}-\mathbb{P}_{i,i+1})+2g^4\sum_{i=1}^L (\mathbb{I}-\mathbb{P}_{i,i+2})
\end{equation}
which is the Dilatation operator of $\mathcal{N}=4$ SYM up to two loops \cite{BDS}, see (\ref{dilatation0}). Let us summarize the outcome of these last sections. We proved that

\textit{The states (\ref{closed}), (\ref{BBOmega}) are the precise two loop eigenvectors of the two loop Dilatation operator (\ref{dilatation0}) of $\mathcal{N}=4$ SYM. In particular, all the monstrous contact terms discussed in section \ref{contactSec} and appendix \ref{C4} are automatically incorporated in the algebraic approach (\ref{closed}) using the $\Theta$--morphism!}

This is one of the main results of this paper.  Higher loops will be briefly touched upon in section \ref{Axiomatics}. In section \ref{ThetaC123} we will apply these results to the computation of structure constants at one loop level.

\subsection{$\Theta$--morphism and the All Loop Dispersion} \la{Axiomatics}
In this section we will briefly discuss the $\Theta$--morphism from a more axiomatic point of view. The goal is to find out how we can constrain the $\Theta$--morphism action at higher loops. We start by an ansatz of the form
\begin{equation}
\ckappa{\theta}{}{f(\theta)}=
\left.
\exp\(\sum_{i=1}^L\sum_{l=1}^{\infty} g^{2l} P_{2l}(\d_i,\dots,\d_{i+l})\)f(\theta)\right |_{\theta_i=0} \la{locality}
\end{equation}
where $P_{2l}$ is polynomial of degree $2l$.
In what follows we assume that $P_{2l}$ is a homogeneous polynomial of degree $2l$.
Note that in the exponent the polynomial only acts on $l$ neighboring sites. We call this axiom the \textit{locality axiom}. It is inspired by the structure of perturbation theory where the interaction range also increases with the perturbative loop order.

Next we require the second axiom:
\begin{equation}
\ckappa{\theta}{}{f_{\rm sym}(\theta)h(\theta)}=
\ckappa{\theta}{}{f_{\rm sym}(\theta)}
\ckappa{\theta}{}{h(\theta)}\;. \la{morpro2}
\end{equation}
which we call the \textit{morphism axiom}. As explained before, this is a very important property which allows us to treat the Bethe roots as numbers, ignoring their dependence on the impurities $\theta_j$. This seems like a very mild requirement but it turns out to be extremely powerful! We can study the constraint (\ref{morpro2}) in perturbation theory. We did it up to order $g^8$ by considering polynomial functions of increasingly higher degree. The result of this brute force analysis is quite surprising. We found that the morphism axiom fixes completely\footnote{The morphism axiom is by no means sufficient to constrain (\ref{locality}) completely. It does not fix the form of the polynomials $P_{2l}$ at higher $l$. However, it does seem to fix the parameters of the polynomials that enter in the computations of $\ckappa{\theta}{}{\theta_j^n}$. This is quite nice because it is exactly what we need to act on the dispersion relation!}
\beqa
\ckappa{\theta}{}{\theta_j^4} &=& 3 \,\,\,\,g^2\ckappa{\theta}{}{\theta_j^2}  = \,3 \,\,\,g^4 \nonumber\\
\ckappa{\theta}{}{\theta_j^6} &=& 10  \,g^{4}\,\ckappa{\theta}{}{\theta_j^2}=10  \,g^{6}  \nonumber\\
\ckappa{\theta}{}{\theta_j^8} &=& 35\, g^{6}\, \ckappa{\theta}{}{\theta_j^2} =35 \, g^8\la{seq}
\eeqa
The action on $\theta_j^2$ is not constrained and amounts to (re)defining the coupling constant so we set it to $g^2$. We can ask the famous
\textit{The On-Line Encyclopedia of Integer Sequences}\footnote{http://oeis.org/ or directly http://oeis.org/search?q=3\%2C+10\%2C+35}
\begin{equation}
\ckappa{\theta}{}{\theta_j^{2n}}= \frac{(2 n)!}{2 (n!)^2}g^n \, \la{goal}
\end{equation}
which are very familiar in this context \cite{BDS,didina}.
We conjecture that (\ref{locality}) and (\ref{morpro2}) imply (\ref{goal}). We could not find a prove of this simple statement. If true, (\ref{goal}) immediately implies something quite remarkable: \textit{At all loop orders}
\begin{equation}
\ckappa{\theta}{}{\log\prod_{i=1}^L\frac{u-\theta_i+\tfrac i2}{u-\theta_i-\tfrac i2}}=
L\log\frac{x(u+\tfrac i2)}{x(u-\tfrac i2)}\;
\end{equation}
where $x(u)$ are the Zhukowsky variables.
We see that the one loop momentum (with impurities) becomes promoted to the all loop momentum! We believe this is a very strong evidence in favor of this approach. We will continue our first principles study of the $\Theta$--morphism elsewhere \cite{GSV}.

\section{$\Theta$--morphism and One Loop Structure Constants} \la{ThetaC123}

In this section we will use the $\Theta$--morphism to derive the one loop structure constants by  promoting the tree level impure structure constants of section \ref{impure}. Section \ref{coorSec} was also devoted to the study of one loop structure constants at one loop. The main differences between the approach in this section and the method used in section \ref{coorSec} are
\begin{itemize}
\item First, here we will use the Algebraic Bethe ansatz approach together with the $\Theta$--morphism whereas in section \ref{coorSec} we followed the Coordinate Bethe ansatz. Of course, at the end of the day, both formalisms will give exactly the same result for physical quantities such as  (the absolute value of) structure constants!

However, in intermediate stages, scalar products in the coordinate and algebraic basis will differ since the kets are  normalized differently. It is convenient to know the relative normalization to be able to compare not only the final results but also the intermediate ones. For example, at one loop we have (\ref{lambda}). By checking a few examples like (\ref{2ptExample}) at two loops, we can find the relative normalization between the two loop coordinate states -- which we denoted by $|{\bf p}\>$ in section \ref{contactSec} -- and the algebraic states  -- which we denoted by $|{\bf u}\>$ in section \ref{algStatesSec}. We find
\begin{equation}
|{\bf u}\> = \mu \,|{\bf p}\> \,, \qquad \mu=\(1- \frac{\gamma({\bf p})^2}{4g^2} \) \prod_{j=1}^N(e^{-ip_j}-1) \prod_{i<j} {\frak f}(p_i,p_j) \la{conv1}
\end{equation}
where (\ref{gamma0}) and (\ref{fcorr}).

Another minor difference is that in the Coordinate Bethe ansatz (sections \ref{contactSec} and \ref{coorSec}) we use mostly the momenta $p_a$ as variables. In the Algebraic Bethe ansatz (sections \ref{algSec} and \ref{secMor1}) we use mostly the Bethe rapidities $u_a$ as our variables. The relation between the two is given by $u_a=u(p_a)$ where (\ref{Smatrix}).
Equivalently
\begin{equation}
e^{ip} = \frac{x(u+i/2)}{x(u-i/2)} \,, \qquad x(u)= \frac{u+\sqrt{u^2-4g^2}}{2g} = \frac{u}{g}-\frac{g}{u} + \dots
\end{equation}
We will also use $x^{\pm}=x(u\pm i/2)$ and $x_a^{\pm}=x(u_a\pm i/2)$.  For example, the above equation (\ref{conv1}) can equivalently be written as
\begin{equation}
\mu=({1- g^2\Gamma_u^2}) \prod_{j=1}^N\(\frac{x_j^-}{x_j^+}-1\) \prod_{i<j} {\frak f}(u_i,u_j) \la{conv2}
\end{equation}
where
\begin{equation}
\Gamma_{\bf u} \equiv \sum_{j=1}^N \frac{1}{u_j^2+\frac{1}{4}}\,, \qquad  {\frak f}(u,v)\equiv \(1+\frac{i}{u-v}\)\(1+\frac{g^2}{(u^2+\frac{1}{4})(v^2+\frac{1}{4})}\)
\end{equation}
The set of momenta of the three operators ${\mathcal O}_1$, ${\mathcal O}_2$ and ${\mathcal O}_3$ is, respectively ${\bf k}$, ${\bf p}$ and ${\bf q}$ while the corresponding rapidities are denoted by ${\bf u}$, $\bf v$ and $\bf w$.

\item The second important difference between the two approaches is in the method used.
In the coordinate approach of section \ref{coorSec} we start with the scalar products for the one loop eigenstates and we \textit{guessed} the higher loop generalizations. These conjectures were based on (1) playing with examples with a small number of excitations and finding patterns and (2) using the experience coming from higher loop corrections in the spectrum problem.
Sometimes, one can guess not only the first quantum correction but even conjecture all loop results, see e.g. section \ref{normsec}. However, for the most complicated expectation value \verb"involved" (\ref{tocompute}) the guessing strategy was manifestly insufficient  to guess even the first quantum correction. With the algebraic Bethe ansatz approach we have full control over the two loop eigenvectors. As such, all one loop structure constants can be derived using the $\Theta$--morphism without any guessing.

In sum, the two methods are nicely complementary.
\end{itemize}

\subsection{Derivation of the Norm}
In (\ref{tocompute}) we identified all required scalar products and expectation values that we need to compute to obtain the complete one loop corrected structure constants in the $SU(2)$ setup of \cite{paper1}. We will now compute them using the algebraic Bethe ansatz approach.

In this section we consider the simplest ones, the norms $\<{\bf 1}|{\bf 1}\>$ etc. The norms were studied in the coordinate Bethe ansatz language in section \ref{normsec}. In the algebraic Bethe ansatz the corrected two loop eigenvectors are given by (\ref{closed}), $|{\bf 1}\> = |{\bf u}\>$ so what
we want to compute is
\begin{equation}\la{closednorm}
\langle{\bf u}|{\bf u}\rangle\equiv
\ckappa{\theta}{}{\langle\theta;{\bf u}|}
(1-2g^2 \Gamma_u {\mathbb H}_{L,1})
\ckappa{\theta}{}{|\theta;{\bf u}\rangle}\;.
\end{equation}
Note that for cyclic states the insertion of the operator ${\mathbb H}$
will not depend of the position where it is inserted.
This observation allows for the replacement ${\mathbb H}_{L,1}\to \Gamma_{\bf u}/L$ since we can average over where we insert. We will not need to use this in the derivation.

The strategy is now to convert the contraction of two $\Theta$-deformed
states to the overall $\Theta$-deformation of the scalar product
\begin{equation}\la{Ntheta}
\ckappa{\theta}{}{\langle\theta;{\bf u}|\theta;{\bf u}\rangle}\;.
\end{equation}
Note that we do know the explicit expression for the quantity inside the brackets; it is given in (\ref{normB}). Hence evaluating (\ref{Ntheta}) is straightforward (we will do it below).

To bring (\ref{Ntheta}) to a form closer to (\ref{closednorm}) we have to compute the cross-term. Using the identities derived in section \ref{IdSec}, in particular (\ref{PsiId}), we easily find
\beqa
\cckappa{\theta}{}{\langle\theta;{\bf u}|}{|\theta;{\bf u}\rangle}
&=&\sum_{j=1}^L g^2\langle 0;{\bf u}|({\mathbb H}_{j,j+1}-\delta_{j,L}\Gamma_{\bf u})^2|0;{\bf u}\rangle\\
&=&g^2\langle 0;{\bf u}|2\Gamma_{\bf u}+\Gamma_{\bf u}^2-2\Gamma_{\bf u}{\mathbb H}_{1,L}|0;{\bf u}\rangle\;. \la{this}
\eeqa
We see that the term with insertion of the Hamiltonian density is exactly the same as in (\ref{closednorm})!\footnote{In (\ref{this}) we can replace $|0;{\bf u}\rangle$ by $\ckappa{\theta}{}{|\theta;{\bf u}\rangle}$ if we want to. At this loop order they are indistinguishable since (\ref{this}) already has an overall $g^2$ factor.}
Hence
\begin{equation}
\langle{\bf u}|{\bf u}\rangle =(1-2g^2 \Gamma_{\bf u}-g^2 \Gamma_{\bf u}^2) \ckappa{\theta}{}{\langle\theta;{\bf u}|\theta;{\bf u}\rangle}\;, \la{normPart1}
\end{equation}
which is exactly the kind of expression we were after. Furthermore, as mentioned above, (\ref{Ntheta}) can be readily computed: the function (\ref{normB}) only depends on the impurities $\theta_j$ through the symmetric combination appearing in (\ref{disppro}). Hence, using (\ref{disppro}) and (\ref{lema}) we immediately conclude that all we need to do is replace the dispersion factor, that is
\begin{equation}
 \ckappa{\theta}{}{\langle\theta;{\bf u}|\theta;{\bf u}\rangle}=\prod_{m\neq k} \frac{u_k-u_m+i}{u_k-u_m}   \det_{j,k\le N_1} \frac{\partial }{\partial u_j} \[ \frac{L}{i}  \log \frac{x(u_k+i/2)}{x(u_k-i/2)}
+\frac{1}{i}\sum_{m\neq k}^{N_1} \log\frac{u_k-u_m-i}{u_k-u_m+i}\]   \nonumber
\end{equation}
Taking into account the conversion factor (\ref{conv1}) one can check that this is in perfect agreement with the conjecture for the norm of section \ref{normsec}!

\subsection{Derivation of the DVD Amplitude}
Having computed the norms we now move to the next line in (\ref{tocompute}), the contribution \verb"simple". As explained in section \ref{simplificationSec} this quantity can be simplified further so that all we need is the scalar product (\ref{Adef}) between a Bethe state and a state obtained by gluing together two different vacuum descendents.

For one loop eigenvectors with impurities (\ref{BBOmega}) this quantity is given in  (\ref{tree132}).
On the other hand, we are interested in the two loop eigenvectors with no impurities (\ref{closed}), that is
$
\<\text{vac}| {\bf v} \>
$.
To relate the two we follow the same strategy as in the previous section and start by applying the $\Theta$--morphism to (\ref{tree132}). We get
\begin{equation}
\ckappa{\theta}{}{\<\text{vac}|\theta^{(2)} ; {\bf v}\rangle}=
\<\text{vac}| \ckappa{\theta}{}{|\theta^{(2)} ; {\bf v}\rangle}=\<\text{vac}|{\bf v} \>+
g^2 \Gamma_{\bf v}\, \<\text{vac}| {\mathbb H}_{1,L_2} |0;{\bf v} \rangle\;.
\end{equation}
The first term in the right hand side is exactly what we are after. As for the last one we can use (\ref{PsiId}) with $j=L=L_2$ to get
\begin{equation}
\<\text{vac}|{\bf v} \>=\ckappa{\theta}{}{\<\text{vac}|\theta^{(2)} ; {\bf v}\rangle}-
g^2 \Gamma_{\bf v}  (i\partial_{1}-i\partial_{L_2} + \Gamma_{\bf v})\, \<\text{vac}|
\theta^{(2)};{\bf v} \rangle \la{almost}
\end{equation}
Note that in the right hand side of (\ref{tree132}) there is no dependence on $\theta_{L_2}$ and thus we can drop the $\partial_{L_2}$ from (\ref{almost}). All we have to do now is compute the derivatives in the right hand side of (\ref{almost}).
For that we use
\beqa
\ckappa{\theta}{}{{ e}^{L_{12}}_{\bar \alpha} } =(1-g^2 \Gamma_{\bar \alpha}^2) {\frak e}^{L_{12}}_{\bar \alpha}+ \mathcal{O}(g^4)\, , \qquad  g^2 i \partial_1 \({ e}^{L_{12}}_{\bar \alpha}  \) = -g^2 \Gamma_{\bar \alpha}{\frak e}^{L_{12}}_{\bar \alpha} + \mathcal{O}(g^4)
\eeqa
where we use the short-hand notation
\begin{equation}
{\frak e}^{L_{12}}_{\bar \alpha} \equiv \prod_{\bar a \in \bar\alpha} \(\frac{x(\bar a+i/2)}{x(\bar a-i/2)}\)^{L_{12}}\;.
\end{equation}
As usual, the $\Theta$--morphism naturally gives rise to the Zhukowsky variables. The obtained result
$\<\text{vac}|{\bf v} \>=(1- {g^2} \Gamma_{\bf v}^2)
\sum\limits_{\alpha \cup \bar\alpha=\bf v}
(-1)^{|\bar\alpha|}
f^{\alpha,\bar\alpha}{\frak e}^{L_{12}}_{\bar\alpha}\(1+{g^2} \Gamma_{\alpha}\Gamma_{\bar\alpha}\) $ can be simplified further to
\beqa
\<\text{vac}|{\bf v} \>
&=&(1- {g^2} \Gamma_{\bf v}^2)\sum_{\alpha \cup \bar\alpha=\bf v}
(-1)^{|\bar\alpha|}
{\frak f}^{\alpha,\bar\alpha}{\frak e}^{L_{12}}_{\bar\alpha} \,, \la{Aalg}
\eeqa
if we introduce a modified function
\begin{equation}
{\frak f}(u,v)\equiv \(1+\frac{i}{u-v}\)\(1+\frac{g^2}{(u^2+\tfrac14)(v^2+\tfrac14)}\)
\end{equation}
to absorb the partial energies $\Gamma_{\alpha}$ and $\Gamma_{\bar \alpha}$. The result we just \textit{derived} coincides precisely with the \textit{conjecture} in section \ref{Asec} based on the coordinate Bethe ansatz approach!\footnote{Of course, to compare both one needs to take (\ref{conv1}) into account.}

Using (\ref{Aalg}), (\ref{normPart1}), (\ref{BAEimpure2})  we can write the final result (\ref{C123oxo}) in the
following succinct form
\begin{equation}
 C_{123}^{\circ\bullet\circ} =
\frac{\displaystyle\sqrt{L_1L_2L_3 }}{\displaystyle\sqrt{\binom{L_1}{N_1}\binom{L_2}{N_3} }}
\frac{\displaystyle(1-{g^2} \Gamma_{\bf v}^2/2)\sum_{\alpha\cup\bar\alpha=\bf v}
(-1)^{|\bar\alpha|}
{\frak f}^{\alpha,\bar\alpha}{\frak e}^{L_{12}}_{\bar\alpha}}{\displaystyle\sqrt{ \prod_{i\neq j} f(u_i,u_j) \det_{j,k} \partial_{v_j} \phi_k^{\bf v}}}
\;,
\end{equation}
which agrees perfectly with the previously conjectured expression in the coordinate Bethe ansatz approach of section \ref{coorSec}. The case $ C_{123}^{\circ\bullet\bullet}$ is not harder to get, see \ref{decaysec}.

\subsection{Two non-BPS Three Point Function $C_{123}^{\bullet\bullet\circ}$  and also $C_{123}^{\bullet\bullet\bullet}$ }\la{sec73}
In this section we apply the  technology developed above
to the most complicated case of the structure constant when
all three operators are non-BPS. The operator ${\cal O}_3$ can be replaced by the corresponding BPS
operator without any loss of generality since its contribution factorizes and appears only in the \verb"simple" contribution in (\ref{tocompute}).
 Hence, we will first consider this to be the case and
 recover the general case at the very end.

Our starting point is again \eq{closed}. We pick two closed states at two loops
\beqa\la{closed2}
|{\bf 1}\rangle&\equiv&
(1-g^2 \Gamma_{\bf u} {\mathbb H}_{L_1,1})
\ckappa{\theta^{(1)}}{ }{|\theta^{(1)};{\bf u}\rangle} \\
|{\bf 2}\rangle&\equiv&
(1-g^2 \Gamma_{\bf v} {\mathbb H}_{L_2,1})
\ckappa{\theta^{(2)}}{ }{|\theta^{(2)};{\bf v}\rangle} \nonumber
\eeqa
Then we have to compute the most complicated contribution in (\ref{tocompute}) which is the last one, denoted as \verb"involved". We have
\begin{equation}
\verb"involved"=\<{\bf 1}|\hat{\mathcal{O}}_3|{\bf 2}\> -g^2 \<{\bf 1} | \(\mathbb{H}_{L_{12},L_{12}+1}+\mathbb{H}_{L_1,1}  \) \hat{{\mathcal O}}_3 \(\mathbb{H}_{L_{12},L_{12}+1}+\mathbb{H}_{L_2,1}  \) |{\bf 2}\>  \la{homework}
\end{equation}
where
\begin{equation}
\hat{{\mathcal O}}_3 = \sum_{i_1=\uparrow,\downarrow} \dots \sum_{i_{L_{12}}=\uparrow,\downarrow} \Big|i_1 \dots i_{L_{12}} \underbrace{\downarrow \dots \downarrow}_{L_{13}=N_3} \Big\> \Big\< i_1 \dots i_{L_{12}} \underbrace{\uparrow \dots \uparrow}_{L_{23}=L_3-N_3} \!\!\Big|  \,.
\end{equation}
It is also convenient to define the one loop states as $|{1}\> \equiv |0,{\bf u}\> $ and similar for $|{ 2}\>$. We shall not compute the second term in (\ref{homework}) since we will see soon that it neatly cancels with a contribution from the first one. Hence we focus on simplifying that first term
\beqa\la{startCxxo}
\<{\bf 1}|\hat{\mathcal{O}}_3|{\bf 2}\>&=&
\ckappa{\theta^{(1)}}{}{\langle\theta^{(1)};{\bf u}|}
\ckappa{\theta^{(2)}}{}{\hat O_3|\theta^{(2)};{\bf v}\rangle}
-g^2\langle 1|
( \Gamma_{{\bf u}} {\mathbb H}_{L_1,1}\hat O_3
+\Gamma_{\bf v} \hat O_3 {\mathbb H}_{L_2,1})
|2 \rangle\;.
\eeqa
Again we have the $\Theta$--morphism acting on the bra and ket states separately.
The strategy is as before: we want to to bring it to the form of
a differential operator in $\theta$'s acting on the tree level expression with impurities
\begin{equation}
 \<\theta^{(1)} ;{\bf u} | \hat{\mathcal{O}}_3| \theta^{(2)};{\bf v} \> \la{theguy} \,.
\end{equation}
This time it may seem a bit ambiguous since there are two different sets of $\theta$'s involved.
However, as explained in figure \ref{figC123}, the scalar product (\ref{theguy}) only depends on $\theta^{(1)}$ so that it is rather natural
to pick this set of $\theta$'s to act on. Moreover, the argument in that figure actually implies something slightly stronger, namely that the bottom part of the figure does not depend on $\theta^{(23)}$ and hence it only depends on $\theta^{(12)} \subset \theta^{(1)}$. The bottom is nothing but $\hat{{\mathcal O}}_{3} |\theta^{(2)};{\bf v}\rangle$ so we conclude that in (\ref{startCxxo}) the first term is\footnote{Very explicitly:
\beqa
\nonumber\ckappa{\theta^{(1)}}{}{\hat O_3|\theta^{(2)};{\bf v}\rangle}
&=&
\hat O_3|{\bf v}\rangle+\sum_{i=1}^{L_1}\frac{g^2}{2}\(\d_{\theta^{(1)}_{i}}-\d_{\theta^{(1)}_{i+1}}\)^2 {\hat O_3|
\theta^{(2)};{\bf v}\rangle}
\\
&=&
\hat O_3|{\bf v}\rangle+\frac{g^2}{2}\[
\sum_{i=1}^{L_{12}-1}\(\d_{\theta_{i}^{(1)}}-\d_{\theta^{(1)}_{i+1}}\)^2
+ \d_{\theta^{(1)}_{L_{12}}}^2
+ \d_{\theta^{(1)}_{1}}^2
\]
\ckappa{\theta^{(1)}}{}{\hat O_3|\theta^{(2)};{\bf v}\rangle}
=
\nonumber\ckappa{\theta^{(2)}}{}{\hat O_3|\theta^{(2)};{\bf v}\rangle}\;.
\eeqa}
\beqa\la{ourgoal}
\ckappa{\theta^{(1)}}{}{\langle\theta^{(1)};{\bf u}|}
\ckappa{\theta^{(2)}}{}{\hat O_3|\theta^{(2)};{\bf v}\rangle} = \ckappa{\theta^{(1)}}{}{\langle\theta^{(1)};{\bf u}|}
\ckappa{\theta^{(1)}}{}{\hat O_3|\theta^{(2)};{\bf v}\rangle}
\eeqa
By acting with the $\Theta$--morphism on (\ref{theguy}) we obtain the  right hand side of this expression plus crossed terms that need to be massaged as usual. This is done in detail in appendix \ref{derivationC123}. At the end of the day we find
\beqa\nonumber
\<{\bf 1}|\hat{\mathcal{O}}_3|{\bf 2}\>&=&\ckappa{\theta^{(1)}}{}{\<\theta^{(1)} ;{\bf u} | \hat{\mathcal{O}}_3| \theta^{(2)};{\bf v} \>}+
g^2(\Gamma_{\bf v}-\Gamma_{\bf u})
\[i{\cal D}^{(1)}_{L_{12}}-\Gamma_{\bf v}+\Gamma_{\bf u}\]
\langle \theta^{(1)} ,{\bf u}|\nonumber\hat O_3|\theta^{(2)},{\bf v}\rangle
\\
&-&
g^2 (\Gamma_{\bf u}+\Gamma_{\bf v}
\Gamma_{\bf u}
+\Gamma_{\bf v})\langle{1}|
\hat O_3
|{2}\rangle \nonumber \\
&+&g^2\langle{1}|
{\mathbb H}_{L_{12},1}\hat O_3-
{\mathbb H}_{L_1,1}\hat O_3-\hat O_3
{\mathbb H}_{L_{12},L_{12}+1}-
\hat O_3
{\mathbb H}_{L_2,1}|{2}\rangle\la{cc132sec}\;.
\eeqa
We see that the last line cancels precisely with the second term in (\ref{homework})! Recall that these terms came from internal Feynman loop contributions. We see that they neatly cancel with particular contributions coming from the wave function corrections.

More remarkable simplifications arise when we combine \verb"involved" with the other contributions in (\ref{tocompute}) to obtain the physical quantity $|C^{\bullet\bullet\circ}_{123}|$. For ${\mathcal O}_3$ BPS we have
\begin{equation}
|C^{\bullet\bullet\circ}_{123}|= \sqrt{\frac{L_1L_2L_3}{\binom{L_3}{N_3}}}\times |\verb"involved"/ \sqrt{\<{\bf 1}| {\bf 1}\>\<{\bf 2}| {\bf 2}\>}|
\end{equation}
Then dividing \eq{cc132sec} by the norms of  $|{\bf 1}\>$ and $|{\bf 2}\>$ given by \eq{normPart1} we get
\beqa
\la{almostthere}
\verb"involved"/ \sqrt{\<{\bf 1}| {\bf 1}\>\<{\bf 2}| {\bf 2}\>}&=&
\frac{
\ckappa{\theta^{(1)}}{}{\<\theta^{(1)} ;{\bf u} | \hat{\mathcal{O}}_3| \theta^{(2)};{\bf v} \>}
}{\sqrt{
\ckappa{\theta^{(1)}}{}{\langle\theta^{(1)};{\bf u}|\theta^{(1)};{\bf u}\rangle}
\ckappa{\theta^{(2)}}{}{\langle\theta^{(2)};{\bf v}|\theta^{(2)};{\bf v}\rangle}
}
}\\
&+&\frac{g^2(\Gamma_{\bf v}-\Gamma_{\bf u})}{\sqrt{\<1|1\>\<2|2\>}}
\[i{\cal D}^{(1)}_{L_{12}}-\tfrac{1}{2}\(\Gamma_{\bf v}-\Gamma_{\bf u}\)\]
\<\theta^{(1)} ;{\bf u} | \hat{\mathcal{O}}_3| \theta^{(2)};{\bf v} \> \nonumber
\;.
\eeqa
As we show in Appendix \ref{impart} the last line is purely imaginary and thus
can be absorbed into a physically irrelevant phase prefactor in the left-hand side!
We conclude that the properly normalized structure constant is given by
\begin{equation}
|C^{\bullet\bullet\circ}_{123}|=\left|
\frac{\sqrt{L_1 L_2 L_3}
\ckappa{\theta^{(1)}}{}{\<\theta^{(1)} ;{\bf u} | \hat{\mathcal{O}}_3| \theta^{(2)};{\bf v} \>}
}{\sqrt{\binom{L_3}{N_3}
\ckappa{\theta^{(1)}}{}{\langle\theta^{(1)};{\bf u}|\theta^{(1)};{\bf u}\rangle}
\ckappa{\theta^{(2)}}{}{\langle\theta^{(2)};{\bf v}|\theta^{(2)};{\bf v}\rangle}
}
}
\right|\;.
\end{equation}
In sum, all ugly factors miraculously canceled out leaving us with the nicest possible expression!

From the discussion in section \ref{setup} we can easily generalize to the three BPS case
which only differs by an additional factor corresponding to the third operator:
\begin{equation}
|C^{\bullet\bullet\bullet}_{123}|=\left|
\frac{\sqrt{L_1 L_2 L_3}
\ckappa{\theta^{(1)}}{}{\<\theta^{(1)} ;{\bf u} | \hat{\mathcal{O}}_3| \theta^{(2)};{\bf v} \>}
\mathcal{A}_{N_3}({\bf w})(1-g^2\Gamma_{\bf w}/2)
}{\sqrt{
\ckappa{\theta^{(1)}}{}{\langle\theta^{(1)};{\bf u}|\theta^{(1)};{\bf u}\rangle}
\ckappa{\theta^{(2)}}{}{\langle\theta^{(2)};{\bf v}|\theta^{(2)};{\bf v}\rangle}
\<{\bf 3}|{\bf 3}\>
}
}
\right|\;. \la{super}
\end{equation}

Finally, it is straightforward to compute all derivatives appearing in the several $\Theta$--morphisms since we know the impure results, see section \ref{impure}. The final result for the one loop structure constants is presented in Appendix \ref{fullAfter}. The expression (\ref{super}) -- or equivalently the more explicit expressions in the Appendix \ref{fullAfter} -- is the main practical outcome of the $\Theta$--morphism.

\section{Conclusions} \la{Conc}
Let us conclude with a small list of the main ideas/results together with a link to the corresponding equations.

We constructed the eigenvectors of the two loop dilatation operator (\ref{dilatation0}) both in Coordinate
and Algebraic Bethe ansatz formalisms.
Using the Coordinate Bethe ansatz this amounts to finding all contact terms, see (\ref{contactTerms}) and (\ref{ctA}). In the Algebraic Bethe Ansatz approach we used the $\Theta$--morphism (\ref{mor1}) to promote the one loop eigenvectors
with extra parameters (impurities)
 (\ref{stateAlg}) into exact two loop
eigenvectors (\ref{closed}). In (\ref{Norm}) we conjectured the form of the norm of the states at {\it any loop order}
and made various tests of the conjectures by considering several states up to four loops.
We also give a derivation using the algebraic approach at two loops  in (\ref{normPart1}).

Another important scalar product is the Double Vacuum Decay (DVD) amplitude (\ref{Adef}),
which we found up to two loops in (\ref{Afin}) and (\ref{almost}). Using DVD as a building block
we found the structure constants $C_{123}^{\circ\bullet\bullet}$ (for ${\mathcal O}_1$ BPS)
 in (\ref{C123oxx}).

Finally, the $\Theta$--morphism techniques were used to derive the most general
structure constants $C_{123}^{\bullet\bullet\bullet}$ (all are non-BPS) in
  (\ref{super}) and (\ref{finalEvaluated}).

{  These expressions are basically ratios of  simple determinants.
As such they can be easily evaluated in a computer, even for operators with a large number of magnons which are the relevant operators in the so called classical limit.
 This allows for nice comparisons with strong coupling which were discussed in the note \cite{short}. }

It seems relatively straightforward to further correct the $\Theta$--morphism and take into account in this way the all loop corrections to the dispersion relation \cite{GSV}. In particular, we have full control over finite $L$ states. On the other hand, it seems harder to incorporate the dressing factor. It would be very interesting to make contact with the boost operator approach of \cite{boost} where the dressing factor corrections are under control for large $L$ states. To gain further insight into the underlying structure it is also very important to consider more general scalar products with fermions and derivatives.
We hope that with some more data, a unifying structure will emerge and very general all loop
expressions will become accessible.

\subsection*{Acknowledgments}
We thank J. Caetano, J. Escobedo, O. Foda, I. Kostov, D. Serban, A. Sever, J. Toledo, S. Valatka, T. Wang for discussions and suggestions.
N.G. (P.V.) would like to thank Nordita and the Perimeter Institute (King's College London, IHP and Nordita) for warm hospitality.
Research at the Perimeter Institute is supported in part by the Government of Canada through NSERC and by the Province of Ontario through MRI.

\newpage

redpajama_set_name: RedPajamaArXiv


text:
Achievements of FLP
State Services
FLP Manifesto for the 2018 General Elections.
Australian Unionists Call For Court's Constitutional Decision to be Implemented
// Display comment count + link
Following the Fiji Supreme Court's decision that the Qarase Government's exclusion of the Fiji Labor Party (FLP) from Government was unconstitutional and illegal, visiting Australian unionist have joined calls for the Chaudhry led FLP to take its rightful place in Government, including calling on the Australian Government to intervene in facilitating the decision of the Supreme Court.
"This decision of the Fiji Supreme Court is a fantastic step forward and a vindication of the FLP's fight for justice and democracy in Fiji", head of the NSW Branch of the Australian Transport Worker's Union Tony Sheldon said today.
"With the court's vindication and instruction, the Qarase Government must move to immediately comply with the requirements of the 1997 Constitution and implement the participation of Mahendra Chaudhry's FLP in Government."
"The court's decision is clear, and Australian unions expect the Government to make the right decision," Australian Health Services Union National President Michael Williamson said.
"A further deterioration into the crisis, instability and uncertainty that has riddled Fiji since the illegal coup in May 2000 is in nobody's interests and our organizations' will be doing everything we can to make sure that this doesn't happen."
"Including, immediately calling on the Australian Government to review all aid, assistance and favorable trading arrangements to Fiji and Australian businesses in Fiji unless the courts decision is implemented," Australian NSW Textile Clothing and Footwear Secretary Barry Tubner said.
The Australian union delegation is visiting Fiji to attend the Fijian Labour Party Conference in Nadi this weekend Friday 24th and 25th of July.
FLP Delegates Conference Discuss Multi Party Issue
The FLP Delegates Conference was today briefed by the FLP parliamentary leader, Mahendra Chaudhry, on his discussions with Prime Minister Laisenia Qarase on the invitation…read more →
Qarase's attack un-prime minister-like: Chaudhry
Labour leader Mahendra Chaudhry has rubbished a bitter personal attack launched on him by Laisenia Qarase as "unbecoming of a prime minister''. Qarase on Friday…read more →
Chaudhry meets Qarase on national issues
Labour Leader Mahendra Chaudhry met with Prime Minister Laisenia Qarase today in the first of their one-to-one talks to break the ice between the country's…read more →
Land as a vital resource
"As owners of the nation's major resources – land, forests, fisheries and minerals – it is unconscionable, indeed tragic, that the indigenous people have largely remained poor and their resources largely undeveloped": Mahendra Chaudhry It is worth pondering why despite....
Ba Province wants an effective voice in governance
"In my view, successive governments could and should have done much more to promote the interests and welfare of the iTaukei community.
Bird's eyeview of the Namaka Market
The new $5.3m Namaka market complex in Nadi has a serious issue with bird poop from nesting pigeons posing a health hazard and affecting businesses located at the market, complain vendors and operators.
Quo vadis Sugar?
The current controversy surrounding the final cane payment for the 2019 season vis a vis the government promised guaranteed price of $85 per tonne, will further erode growers' confidence in the sugar industry: Mahendra Chaudhry In a statement issued on....
Human trafficking and sex racketeering in Fiji
Two recently published reports reveal a shocking picture of the existence of a well organized racket in Fiji that commercially exploit women and children in sexual activities ranging from prostitution, pornography and sex trafficking. The first, is a US Government....
Looking Behind the News:
Digging Deeper into Fiji Airways
Fiji Airways Chief Executive Andre Viljoen deserves a rap on the knuckles for his comment that the airline does not respond to political statements, says Labour Leader Mahendra Chaudhry.
Keeping FNPF in check
The Fund is a superannuation Trust worth billions in assets and members' savings.
Has the FNPF lost its way?
"There are genuine concerns about a lack of transparency in the disclosures pertaining to the financials of the Fund's associated companies and its subsidiaries": Mahendra Chaudhry A full page ad by FNPF (FT 1 July 2020) proudly announcing a 5% interest....
Time for a hard look at Fiji Airways
Fiji Airways is in serious trouble, there is no doubt about that.
Fiji "is in a deep mess": Fiji Labour Party
Labour Leader Mahendra Chaudhry told the Party's Annual Delegates Conference in Lautoka today that the government's financial situation was so critical, it was not in a position to pay its creditors.
Opinion and Reports
Draconian Decrees
With humility and honesty … we must serve the people of the land. That challenge is for all of us. National development plans must ensure that the interests of the masses become paramount. If our ordinary people lose faith through unfair and inequitable policies, then the future is fraught with uncertainties. We must restore the faith of our people in democracy, in the government of the day, and above all in Fiji.
- Dr Timoci Bavadra, FLP Prime Minister -	Read More >>
All Rights Reserved Fiji Labour Party.

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Associate Dean of Student Success. Grow student-centered STEM programming.
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Personal and Family Preparedness for Emergencies.
Some of the things you can do to prepare for the unexpected, such as making an emergency supply kit and developing a family emergency plan, can be used in any type of emergency.
The Cascade Mountain Range harbors many volcanoes. Here are steps you can take during and after an eruption to help ensure the safety of yourself, your family and your property.
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Emergency Management also hosts and conducts emergency management and community preparedness training for emergency management professionals, partners and the public.
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This is a list of Alice Busch's education and training certifications.
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The basic plan is the core document upon which other annexes are built.
Alerts and locations for cooling centers in Multnomah County.


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text:
About 4.5 lakh lorries to be off the roads from Dec 27
About 4.5 lakh lorries in Tamil Nadu would be off roads indefinitely from December 27 to protest against what is seen as overpriced accessories like GPS system and speed governors.
After the verdict of the Madurai bench of the Madras High Court to install speed regulators and reflective tapes on vehicles, the state government has been forcing lorry owners to buy the accessories from particular companies which were overcharging, said MR Kumarasamy of Lorry Owners Federation of Tamil Nadu. He was speaking to reporters on the sidelines of a meeting with lorry owners of various districts on Wednesday.
Alleging that the government was forcing the lorry owners to buy the speed governors sold at ₹1,500 in neighbouring Karnataka but priced at ₹8,000 in Tamil Nadu, he said.
Besides, he said, the owners were being forced to buy GPS units and reflective stickers from particular companies, leading to incurring loss of crores of rupees to the owners. There are 49 approved companies who market these units and stickers at a lesser price than those companies recommended by the government, he said.
Other problems include the capacity of load to be carried by the lorries, penalising the vehicles parked at petrol pumps and corruption at transport checkposts, he said.
In the wake of these issues, the lorry owners have decided to keep their vehicles off the road till the demands are met, he added.