product structure described in the previous subsection. The symmetry algebra for the ${{\mathfrak{su}}}(1,1|2)$ sector implies that for even n, the length of the spin chain is unchanged, and for odd n, interactions change the length of the spin chain by one.

Feynman rules restrict to parity (or charge conjugation) even interactions. In terms of the spin chain, a parity even generator satisfies, for arbitrary fields $X_j$, $${{\mathfrak{J}}}{|X_1 \ldots X_i\rangle} = {|X^\prime_1 \dots X^\prime_f\rangle} \Rightarrow  (-1)^{i+f_i(f_i-1)/2} {{\mathfrak{J}}}{|X_i \ldots X_1\rangle} = (-1)^{f+f_f(f_f-1)/2}{|X^\prime_f \dots X^\prime_i\rangle}$$ where $f_i$ and $f_f$ are the number of fermions in the initial and final states.

Finally, we use the normalization for the coupling constant $$g^2=\frac{{g_{\scriptscriptstyle\mathrm{YM}}}^2 N}{8\pi^2},$$ where $N$ is the rank of the gauge group.

The $\mathcal{O}(g^1)$ solution
-------------------------------

Applying the rules from the previous subsection, at $\mathcal{O}(g^1)$, the corrections will entail replacing one spin site with two, or vice-versa. For instance, a one-site to two-site generator ${{\mathfrak{T}}}$ would act on a generic state as: $$\begin{gathered}
 {{\mathfrak{T}}}{|X_1 \ldots X_i \ldots X_n\rangle} = \notag \\ 
\sum_{a, \, b} c^{X_1}_{ab} {|Y_a Y_b X_2 \ldots X_i \ldots X_n\rangle} + \cdots + \notag \\
(-1)^{(X_1\ldots X_{i-1}){{\mathfrak{T}}}} \sum_{a,\,b} c^{X_i}_{ab} {|X_1 \ldots X_{i-1} Y_a Y_b X_{i+1} \ldots X_n\rangle} + \cdots, \end{gathered}$$ where $Y_a$ and 