Tailoring three-point functions and integrability IV. Θ-morphism

We compute structure constants in N = 4 SYM at one loop using Integrability. This requires having full control over the two loop eigenvectors of the dilatation operator for operators of arbitrary size. To achieve this, we develop an algebraic description called the Θ-morphism. In this approach we introduce impurities at each spin chain site, act with particular differential operators on the standard algebraic Bethe ansatz vectors and generate in this way higher loop eigenvectors. The final results for the structure constants take a surprisingly simple form, recently reported by us in the short note arXiv:1202.4103 . These are based on the tree level and one loop patterns together and also on some higher loop experiments involving simple operators.