Abstract
We present the detailed computation of full-color three-loop three-point form factors of both the stress-tensor supermultiplet and a length-three BPS operator in \U0001d4a9 = 4 SYM. The integrands are constructed based on the color-kinematics (CK) duality and generalized unitarity method. An interesting observation is that the CK-dual integrands contain a large number of free parameters. We discuss the origin of these free parameters in detail and check that they cancel in the simplified integrands. We further perform the numerical evaluation of the integrals at a special kinematics point using public packages FIESTA and pySecDec based on the sector-decomposition approach. We find that the numerical computation can be significantly simplified by expressing the integrals in terms of uniformly transcendental basis, although the final three-loop computations still require large computational resources. Having the full-color numerical results, we verify that the non-planar infrared divergences reproduce the non-dipole structures, which firstly appear at three loops. As for the finite remainder functions, we check that the numerical planar remainder for the stress-tensor supermultiplet is consistent with the known result of the bootstrap computation. We also obtain for the first time the numerical results of the three-loop non-planar remainder for the stress-tensor supermultiplet as well as the three-loop remainder for the length-three operator.
Highlights
We present the detailed computation of full-color three-loop three-point form factors of both the stress-tensor supermultiplet and a length-three BPS operator in N = 4 SYM
We summarize the final numerical results of form factors in table 1, which is computed at a special kinematics point s12 = s23 = s13 = −2
We discuss the detailed construction of the full-color three-loop three-point form factors in N = 4 SYM based on the color-kinematics duality and generalized unitarity methods
Summary
We describe the general strategy for constructing loop integrand by colorkinematics duality and unitarity cuts. We give a brief review of color-kinematics duality and describe how to use it to construct an ansatz for the loop integrand of form. We explain how to apply physical constraints to fix the ansatz, including diagrammatic symmetries and unitarity cuts, where a particular emphasis will be on the application of non-planar cuts. Readers are referred to [44, 67, 95] for some further details of the strategy
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