Higgs amplitudes from $\mathcal{N}=4$ super Yang-Mills theory

Higgs plus multi-gluon amplitudes in QCD can be computed in an effective Lagrangian description. In the infinite top-mass limit, an amplitude with a Higgs and $n$ gluons is computed by the form factor of the operator ${\rm Tr}\, F^2$. Up to two loops and for three gluons, its maximally transcendental part is captured entirely by the form factor of the protected stress tensor multiplet operator $\mathcal{T}_2$ in $\mathcal{N}=4$ supersymmetric Yang-Mills theory. The next order correction involves the calculation of the form factor of the higher-dimensional, trilinear operator ${\rm Tr}\, F^3$. We present explicit results at two loops for three gluons, including the subleading transcendental terms derived from a particular descendant of the Konishi operator that contains ${\rm Tr}\, F^3$. These are expressed in terms of a few universal building blocks already identified in earlier calculations. We show that the maximally transcendental part of this quantity, computed in non-supersymmetric Yang-Mills theory, is identical to the form factor of another protected operator, $\mathcal{T}_3$, in the maximally supersymmetric theory. Our results suggest that the maximally transcendental part of Higgs amplitudes in QCD can be entirely computed through $\mathcal{N}=4$ super Yang-Mills.