\begin{array}{l} (\frac{761}{70}, -\frac{138989861}{6174000})^+ \\ (\frac{ 761}{60},-\frac{419501}{16000})^\pm \end{array} \\ \hline \end{array}$

The two-loop ${{\mathfrak{sl}}}(2)$ S-matrix and diffractionless scattering
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We now perform a new two-loop check of the bosonic ${{\mathfrak{sl}}}(2)$ sector Bethe ansatz of [@Staudacher:2004tk]. Instead of only checking anomalous dimension predictions, we also verify a key part of its derivation, the S-matrix. It is straightforward to restrict to the two-excitation ${{\mathfrak{sl}}}(2)$ sector, consisting of states composed only of $\phi^\downarrow$’s and two or fewer derivatives[^9]. We have computed the internal S-matrix as in [@Staudacher:2004tk], which used ideas introduced in [@Bethe:1931hc] and [@Sutherland:1978xx]. A basis for two excitation states is $${|\Psi_{x_1x_2}\rangle}= {|\ldots\phi^\downarrow\stackrel{\stackrel{x_1}{\downarrow}}{{\mathcal{D}}(\phi^\downarrow)}\phi^\downarrow\ldots\phi^\downarrow\stackrel{\stackrel{x_2}{\downarrow}}{{\mathcal{D}}(\phi^\downarrow)}\phi^\downarrow\ldots\rangle},$$ where the derivatives appear at sites $x_1$ and $x_2$ of the spin chain. Then, the Schrödinger equation $$H {|\Psi\rangle} = E {|\Psi\rangle}, \qquad H = g^2 \, {\delta{\mathfrak{D}}}_2+g^4 \, {\delta{\mathfrak{D}}}_4,$$ is solved by the ansatz $$\begin{gathered}
 {|\Psi\rangle} = \sum_{1 \leq x_1 \leq x_2 \leq L} \left(f_-(\delta x, p_i)e^{ip_1x_1+ip_2x_2}+f_+(\delta x, p_i)e^{ip_2x_1+ip_1x_2} \right) {|\Psi_{x_1x_2}\rangle}, \nonumber \\  \delta x = x_2-x_1. \end{gathered}$$ $L$ is the length of the spin chain, and the $p_i$ are the momenta of the excitations which scatter off each other. Since the Hamiltonian is short-ranged and translationally invariant, for large $\delta x$ the solutions of the Schrödinger equation reduce 