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Hidden Analytic Relations for Two-Loop Higgs Amplitudes in QCD

We compute the Higgs plus two-quark and one-gluon amplitudes ($H \rightarrow q \bar{q} g$) and Higgs plus three-gluon amplitudes ($H \rightarrow 3g$) in the Higgs effective theory with a general class of operators. By changing the quadratic Casimir $C_F$ to $C_A$, the maximally transcendental parts of the $H \rightarrow q \bar{q} g$ amplitudes turn out to be equivalent to that of the $H \rightarrow 3g$ amplitudes, which also coincide with the counterparts in ${\cal N}=4$ SYM. This generalizes the so-called maximal transcendentality principle to the Higgs amplitudes with external quark states, thus to the full QCD theory. We further verify that the correspondence applies also to two-loop form factors of more general operators, in both QCD and scalar-YM theory. Another interesting relation is also observed between the planar $H \rightarrow q \bar{q} g$ amplitudes and the minimal density form factors in ${\cal N}=4$ SYM.