^{1}

^{2}

^{1}

^{3}.

We present a closed expression for the octagon form factor which appears as a building block in a class of four-point correlation functions in

The discovery of integrability in the planar

The full spectrum of such operators has been obtained for any value of the gauge coupling applying the integrability techniques related to the thermodynamic Bethe ansatz

The contribution of virtual particles in the spectrum of “heavy” operators (i.e., with large dimensions) is suppressed in the weak coupling limit. This is also the case for the three-point functions of such operators. In the strong coupling limit the virtual particles cannot be neglected anymore, and in the cases amenable to analytical treatment their contribution is expressed in terms of Fredholm determinants

In the computation of the four-point functions of heavy operators by hexagonalization, the virtual particles are not suppressed at weak coupling anymore

A sketch of the octagon

The octagon was expressed in Ref.

In this Letter we report a formal solution for the octagon at finite ’t Hooft coupling

We report here the results and the general logic of the derivation, leaving the proofs to an extended paper. We start in the second section with the representation of the contributions of the virtual particles as Fredholm Pfaffians, outlined in

In this section we recall the series expansion of the octagon as a sum over virtual particles, which will be our starting point. The virtual particles and their bound states propagate in the mirror dynamics and their energy and momentum are written, with the help of the shift operator

Our goal is to evaluate the octagon with four physical and four mirror edges with no excitations at the physical edges, as shown schematically in Fig.

The four-point function depends on the cross ratios in the Minkowski and in the flavor spaces, parametrized in Ref.

The expansion for the octagon, Eq.

To render the Pfaffian representation useful we have to find a way to also perform the multiple integrations over the rapidities in the expansion of

The matrix elements of

In the rest of this Letter we will focus on the weak coupling expansion of the octagon. We will demonstrate the efficiency of our Eq.

The perturbative expansion of the matrix elements of

Substituting this expansion in moments [Eq.

The matrix elements of

Equations

In Ref.

This expansion in the minors of the matrix of ladders

It is straightforward to extract from Eq.

The representations Eq.

The authors are obliged to Till Bargheer, Frank Coronado, and Pedro Vieira for discussions and useful exchanges. This research is partially supported by the Bulgarian NSF Grant No. DN 18/1 and by the bilateral Grant STC/Bulgaria-France 01/6, PHC RILA 2018 No. 38658NG.