ingoing momenta should equal the set of outgoing momenta, and $(iii)$ the $n\to n$ scattering amplitude should factorize into a product of $2\to 2$ scattering amplitude [@Zamolodchikov:1978xm]. This final requirement implies the Yang-Baxter equation, a consistency condition for equivalent orderings of scattering of three-particle states, which can be represented diagrammatically as follows: $$\begin{tikzpicture}[line width = 2pt,scale=0.6,rotate=90,baseline=5.5em,yscale=-1]
\draw[->] (0,{16*(-5+4*sqrt(3))/23}) -- (8,{24*(12-5*sqrt(3))/23});
\node at (-0.4,{16*(-5+4*sqrt(3))/23-0.2}) {{\large $\vartheta_2$}};
\draw[->] ({16*(-5+4*sqrt(3))/23},0) -- ({24*(12-5*sqrt(3))/23},8);
\node at ({16*(-5+4*sqrt(3))/23},-0.45) {{\large $\vartheta_1$}};
\draw[->] ({8*(29-14*sqrt(3))/23},{24*(2+3*sqrt(3))/23}) -- ({24*(2+3*sqrt(3))/23},{8*(29-14*sqrt(3))/23});
\node at ({8*(29-14*sqrt(3))/23-0.3},{24*(2+3*sqrt(3))/23+0.3}) {{\large $\vartheta_3$}};
\node[circle,draw=black,fill=black!02,line width = 2pt,opacity=.90] at ({8*(7-sqrt(3))/23},{8*(7-sqrt(3))/23}) {{\Large $\theta_{12}$}};
\node[circle,draw=black,fill=black!10,line width = 2pt,opacity=.90] at ({8*(19-6*sqrt(3))/23},{8*(16+sqrt(3))/23}) {{\Large $\theta_{13}$}};
\node[circle,draw=black,fill=black!18,line width = 2pt,opacity=.90] at ({8*(16+sqrt(3))/23},{8*(19-6*sqrt(3))/23}) {{\Large $\theta_{23}$}};
\end{tikzpicture}
\qquad
\boldsymbol{=}
\qquad
\begin{tikzpicture}[line width = 2pt,scale=0.6,rotate=90,baseline=21.5em,yscale=-1]
\draw[->] (11,{8-24*(12-5*sqrt(3))/23}) -- (11+8,{8-16*(-5+4*sqrt(3))/23});
\node at (11-0.4,{8-24*(12-5*sqrt(3))/23-0.2}) {{\large $\vartheta_2$}}; 
\draw[->] ({11+8-24*(12-5*sqrt(3))/23},0) -- ({11+8-16*(-5+4*sqrt(3))/23},8);
\node at ({11+8-24*(12-5*sqrt(3))/23},-0.45) {{\large $\vartheta_1$}};
\draw[->] ({11+8-24*(2+3*sqrt(3))/23},{8-8*(29-14*sqrt(3))/23}) -- ({11+8-8*(29-14*sqrt(3))/23},{8-24*(2+3*sqrt(3))/23});
\node at ({11+8-24*(2+3*sqrt(3))/23-0.3},{8-8*(29-14*sqrt(3))/23+0.3}) {{\large $\vartheta_3$}}; 
\node[circle,draw=black,fill=black!02,line width = 2pt,opacity=.90] at ({11+8-8*(7-sqrt(3))/23},{8-8*(7-sqrt(3))/23}) {{\Large $\theta_{12}$}};
\node[circle,draw=black,fill=black!10,line width = 2pt,opacity=.90] at ({11+8-8*(19-6*sqrt(3))/23},{8-8*(16+sqrt(3))/23}) {{\Large $\theta_{13}$}};
\node[circle,draw=black,fill=black!18,line width = 2pt,opacity=.90] at ({11+8-8*(16+sqrt(3))/23},{8-8*(19-6*sqrt(3))/23}) {{\Large $\theta_{23}$}};
\end{tikzpicture}\label{ybe}$$ As we will see in the following sections, for these models the unitarity techniques appear to work well giving the one-loop S-matrix associated to an “integrable quantization” up to finite shifts in the 