All five-loop planar four-point functions of half-BPS operators in $$ \mathcal{N}=4 $$ SYM

Abstract

We obtain all planar four-point correlators of half-BPS operators in N = 4 SYM up to five loops. The ansatz for the integrand is fixed partially by imposing lightcone OPE relations between different correlators. We then fix the integrated correlators by comparing their asymptotic expansions with simple data obtained from integrability. We extract OPE coefficients and find a prediction for the triple wrapping correction of the hexagon form factors, which contributes already at the five-loop order.

Perturbative four-point functions in planar $$ \mathcal{N}=4 $$ SYM From hexagonalization

“…Again we verified it using the OPE expansion provided in the ancillery of[24]. A proof of this identity is in progress by Vasco Goncalves.…”

“…At this order we found a perfect match for both simplest and asymptotic correlators. At four loops we compared with the results of [24] but only in the OPE limit, finding good agreement as well. By doing this latter comparisson we were able to recognize a new identiy between JHEP01(2019)056 Figure 16.…”

“…conformal integrals, 14 see figure 16. Finally, at five loops, our results were already used as input to fix some coefficients in the Ansatz of [24].…”

“…Their results are given as linear combinations of conformal integrals with known coefficients and integrands but in most cases with unknown integrated expressions, specially at four loops and higher. Up to three loops the integrand was found in [23] and the recent paper [24] 1 provides the planar integrand up to five loops. 2 Alternatively hexagonalization [5] 3 provides a non-perturbative method, any value of the 't Hooft coupling, which relies on the integrability of the two-dimensional effective world-sheet of N = 4 SYM.…”

Hexagons and correlators in the fishnet theory

“…Progress with the hexagon formalism is also hindered by the need of renormalising the divergences that show up at wrapping orders [42], when the magnons can circulate around a (non-protected) local operator. To date, no systematic removal of these divergences is known and it is challenging to push the hexagon strategy to higher loops in N = 4 SYM, even for the simplest structure constant, with one non-protected and two half-BPS operators, see [43][44][45]41] for the state of the art on the field theory side.…”