Full-color three-loop three-point form factors in N=4 SYM

Abstract

We present the detailed computation of full-color three-loop three-point form factors of both stress-tensor supermultiplet and a length-three operator in N " 4 SYM. The integrands are constructed based on the color-kinematics 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 after simplifying the integrands. We further perform a numerical evaluation of the integrals at a special kinematics point by using public packages FIESTA and pySecDec based on the sector-decomposition approach. While such a three-loop computation requires large computational resources, we find that one can significantly simplify the numerical computation by expressing the integrands in terms of uniformly transcendental integrals. Given the full-color numerical results, we are able to check the non-planar infrared divergences where the non-dipole terms firstly appear at three loops. The numerical planar finite remainder for the stress-tensor supermultiplet is consistent with the computation from form factor bootstrap. We also obtain for the first time the numerical three-loop non-planar remainder for the stress-tensor supermultiplet, as well as the three-loop remainder for the length-three operator.12 Interestingly, at three-loop level, color factors similar to C a,b in (4.4) contain Nc-subleading parts, such as Nc f a 1 a 2 a 3 . Only after summing up Ca and C b does the subleading parts cancel.13 Note that in (3.14), the summation over σ includes σi P S3{Z3. Here we are only left with σe P S3, since the summation on σi has been taken into account in organizing integrals in pairs to get the N 3 c da 1 a 2 a 3 color factor.

Color-Kinematics Duality and Dual Conformal Symmetry for A Four-loop Form Factor in N=4 SYM

“…Reader is referred to [14,26,51] for further details of the strategy. Here our construction will follow closely the three-loop computation in [45].…”

“…For example, in the first diagram of Figure 3, ℓ a = x 1 − x a ≡ x 1a and the ansatz for the numerator N m 1 is a polynomial of proper distances x 2 ij . Moreover, due to the good UV behavior of the form factor, we can impose the following powercounting constraints on the ansatz: a one-loop n-point sub-graph carries no more than n − 4 powers of the corresponding loop momentum [26], with an exception that if the sub-graph is a one-loop form factor, the maximal power is n − 3 [42], see also [45] for details. As an example, for the first master in Figure 3, monomials like (x 2 ij ) 2 (x 2 ai ) 2 (x 2 dj ) and (x 2 ij ) 3 (x 2 ad )(x 2 aj ), i, j = 1, 2, 3, 4, should be considered, while x b and x c are not allowed to appear.…”

“…We find that the quintuple cuts provide no more constraints on parameters. We emphasize that we have checked both planar and non-planar cuts, and details for performing cuts can be found in [45]. After all these cuts, we end up with a solution with 133 parameters.…”

“…These dual Jacobi relations put strong constraints on the loop integrands, so that by combining the CK-duality with the generalized unitarity method [22][23][24], the construction of full-color loop integrands are made particularly efficient. Such a method has been successfully applied to high-loop amplitudes in SYM [25][26][27][28][29][30][31][32][33][34][35][36][37] and pure YM [38][39][40][41], and also to form factors in N = 4 SYM [16,[42][43][44][45]. In this paper, we extend the applicable scope of the CK duality by obtaining the four-loop solution of the three-point form factor.…”

An Operator Product Expansion for Form Factors III. Finite Coupling and Multi-Particle Contributions

“…Corresponding perturbative data is available in [41][42][43][44][45][46][47][48][49][50][51][52][53][54].…”