Amplitudes, form factors and the dilatation operator in N = 4 $$ \mathcal{N}=4 $$ SYM theory

We study the form factor of a generic gauge-invariant local composite operator in N = 4 $$ \mathcal{N}=4 $$ SYM theory. At tree level and for a minimal number of external on-shell super fields, we find that the form factor precisely yields the spin-chain picture of integrability in the language of scattering amplitudes. Moreover, we compute the cut-constructible part of the one-loop correction to this minimal form factor via generalised unitarity. From its UV divergence, we obtain the complete one-loop dilatation operator of N = 4 $$ \mathcal{N}=4 $$ SYM theory. Thus, we provide a field-theoretic derivation of a relation between the one-loop dilatation operator and the four-point tree-level amplitude which was observed earlier. We also comment on the implications of our findings in the context of integrability