Deciphering the Maximal Transcendentality Principle via Bootstrap

We prove the principle of maximal transcendentality for a class of form factors, including the general two-loop minimal form factors, the two-loop three-point form factor of ${\rm tr}(F^2)$, and the two-loop four-point form factor of ${\rm tr}(F^3)$. Our proof is based on a recently developed bootstrap method using the representation of master integral expansions, together with some unitarity cuts that are universal in general gauge theories. The maximally transcendental parts of the two-loop four-gluon form factor of $\mathrm{tr}(F^3)$ are obtained for the first time in both planar $\mathcal{N}=4$ SYM and pure YM theories. This form factor can be understood as the Higgs-plus-four-gluon amplitudes involving a dimension-seven operator in the Higgs effective theory. In this case, we find that the maximally transcendental part of the $\mathcal{N}=4$ SYM result is different from that of pure YM, and the discrepancy is due to the gluino-loop contributions in $\mathcal{N}=4$ SYM. In contrast, the scalar-loop contributions have no maximally transcendental parts. Thus, the maximal transcendentality principle still holds for the form factor results in $\mathcal{N}=4$ SYM and QCD, after a proper identification of the fundamental quarks and adjoint gluinos as $n_f \rightarrow 4N_c$. This seems to be the first example of the maximally transcendental principle that involves fermion-loop contributions. As another intriguing observation, we find that the four-point form factor of the half-BPS $\mathrm{tr}(\phi^3)$ operator is precisely a building block in the form factor of $\mathrm{tr}(F^3)$.