[Paper Review] Full-color three-loop three-point form factors in N=4 SYM

Guanda Lin, Gang Yang+1 more|arXiv (Cornell University)|Nov 4, 2021

Superconducting Materials and Applications114 references4 citations

TL;DR

This paper presents the first full-color three-loop three-point form factors in $χ=4$ SYM for both the stress-tensor supermultiplet and a length-three BPS operator, using color-kinematics duality and generalized unitarity to construct integrands. It identifies and resolves free parameters in CK-dual integrands, performs numerical evaluations via FIESTA and pySecDec, and confirms non-planar infrared divergences with non-dipole structures at three loops, while providing the first numerical results for three-loop non-planar remainders.

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 N=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.

Motivation & Objective

• To compute full-color three-loop three-point form factors in $χ=4$ SYM for the stress-tensor supermultiplet and a length-three BPS operator.
• To construct integrands using color-kinematics duality and generalized unitarity, addressing the challenge of free parameters in CK-dual representations.
• To numerically evaluate the resulting integrals using sector decomposition via FIESTA and pySecDec, focusing on both planar and non-planar contributions.
• To verify the structure of infrared divergences, particularly the emergence of non-dipole terms at three loops.
• To provide the first numerical results for three-loop non-planar remainder functions for both the stress-tensor supermultiplet and the length-three BPS operator.

Proposed method

• Constructing integrands via color-kinematics duality and generalized unitarity, ensuring consistency between color and kinematic structures at three loops.
• Identifying and resolving a large number of free parameters in CK-dual integrands, showing their cancellation in simplified forms.
• Expressing integrals in a uniformly transcendental basis to simplify numerical evaluation using sector decomposition.
• Performing numerical integration using public tools FIESTA and pySecDec to evaluate both divergent and finite parts of the form factors.
• Comparing numerical results with known bootstrap results for the planar remainder of the stress-tensor supermultiplet.
• Analyzing the structure of infrared divergences, particularly the non-dipole contributions in the non-planar sector.

Experimental results

Research questions

• RQ1 How can full-color three-loop three-point form factors in $χ=4$ SYM be systematically constructed with complete color dependence?
• RQ2 What is the origin and role of the free parameters appearing in CK-dual integrands, and how can they be resolved?
• RQ3 Can numerical evaluation of three-loop form factors be significantly simplified using a uniformly transcendental basis?
• RQ4 Do the non-planar infrared divergences at three loops exhibit non-dipole structures as expected from theory?
• RQ5 What are the first numerical results for the three-loop non-planar remainder functions of the stress-tensor supermultiplet and the length-three BPS operator?

Key findings

• The CK-dual integrands contain a large number of free parameters, which cancel out in the simplified integrand representation, confirming consistency of the construction.
• Numerical evaluation is significantly simplified by expressing integrals in a uniformly transcendental basis, despite the high computational cost of three-loop integrals.
• The non-planar infrared divergences reproduce the expected non-dipole structures, which first appear at three loops.
• The numerical planar remainder for the stress-tensor supermultiplet matches the known result from bootstrap computations, validating the method.
• The paper presents the first numerical results for the three-loop non-planar remainder of the stress-tensor supermultiplet and the three-loop remainder for the length-three BPS operator.
• The non-planar divergent part of the $χ=4$ SYM form factor is found to be $-2(ξ_5 + 2ξ_2ξ_3)/\epsilon$, confirming the structure of non-dipole terms at three loops.