Three-Loop Color-Kinematics Duality: A 24-Dimensional Solution Space Induced by New Generalized Gauge Transformations

We obtain full-color three-loop, three-point form factors of the stress-tensor supermultiplet and also of a length-3 half–Bogomol’nyi-Prasad-Sommerfield operator in $\mathcal{N}=4$ supersymmetric Yang-Mills (SYM) theory based on the color-kinematics duality and on-shell unitarity. The integrand results are verified by all planar and nonplanar unitarity cuts, and they satisfy the minimal power counting of loop momenta and diagrammatic symmetries. Interestingly, these three-loop solutions, while manifesting all dual Jacobi relations, contain a large number of free parameters; in particular, there are 24 free parameters for the form factor of the stress-tensor supermultiplet. Such degrees of freedom are due to a new type of generalized gauge transformation associated with the operator insertion for form factors. We also perform numerical integration and obtain consistent full-color infrared divergences and the known planar remainder. The form factors we obtain can be understood as the $\mathcal{N}=4$ SYM counterparts of three-loop Higgs plus three-gluon amplitudes in QCD and are expected to provide the maximally transcendental parts of the latter.

Figure 1

Trivalent graphs for four-point tree amplitudes.

Figure 2

Loop diagrams related by Jacobi relation.

Figure 3

Trivalent topologies for the form factor of $\mathrm{tr}({\varphi }^2)$.

Figure 4

Master graphs for $\mathrm{tr}({\varphi }^2)$ form factor.

Figure 5

Quadruple cuts for three-loop, three-point form factors.

Figure 6

Trivalent topologies for the form factor of $\mathrm{tr}({\varphi }^3)$.

Figure 7

Master graphs for $\mathrm{tr}({\varphi }^3)$ form factor.

Figure 8

Graph examples of two-loop form factor of $\mathrm{tr}({\varphi }^2)$. (a),(b) Have the same color factor and, after shrinking the propagators labeled by ${\ell }_A$ and ${\ell }_B$, they both reduce to (c).