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We study the form factor of a generic gauge-invariant local compositeoperator in $\mathcal{N}=4$ SYM theory. At tree level and for a minimal numberof external on-shell super fields, we find that the form factor preciselyyields the spin-chain picture of integrability in the language of scatteringamplitudes. Moreover, we compute the cut-constructible part of the one-loopcorrection to this minimal form factor via generalised unitarity. From its UVdivergence, we obtain the complete one-loop dilatation operator of$\mathcal{N}=4$ SYM theory. Thus, we provide a field-theoretic derivation of arelation between the one-loop dilatation operator and the four-point tree-levelamplitude which was observed earlier. We also comment on the implications ofour findings in the context of integrability.

Amplitudes, Form Factors and the Dilatation Operator in $\mathcal{N}=4$ SYM Theory