Stampedes II: Null Polygons in Conformal Gauge Theory

Abstract

We consider correlation functions of single trace operators approaching the cusps of null polygons in a double-scaling limit where so-called cusp times t 2 i = g 2 log x 2 i−1,i log x 2 i,i+1 are held fixed and the t'Hooft coupling is small. With the help of stampedes, symbols and educated guesses, we find that any such correlator can be uniquely fixed through a set of coupled lattice PDEs of Toda type with several intriguing novel features. These results hold for most conformal gauge theories with a large number of colours, including planar N = 4 SYM.

Spinning Hexagons

“…It can be then solved recursively by concatenating the various decoupling poles in each of the rapidities. This is the result (19). From the point of view of the gauge theory, it provides an efficient way of generating tree-level structure constant data for spinning operators of arbitrary twist.…”

“…14 Some corners of the cross ratios space parametrized by the null snowflake OPE limit of the 6-pt function should be controlled by the abelian structure constants. Having complete control over those, we could therefore generalize [12] to the higher point case and provide valuable boundary data for an eventual bootstrap approach to fix these correlators [19,20,21]. limit of the three point functions considered in this paper.…”

“…As derived in (19) the tree-level hexagon partition function can be written as a recursion relation. The minus signs introduced by gradding slightly change the expression (28), but the physics behind it is still the same.…”

“…Beyond tree-level, the hexagon partition function is not a simple rational function of the rapidities. Lack of control over the singularities in other Riemann sheets prevent us from writing a recursion relation akin to (19) at loop order. We are forced to evaluate the scattering of the hexagon form factor via the explicit form of the hexagon partition function (7).…”

“…Crossing can only be performed at finite coupling. In a sense, the simplicity of the tree level result encode some of the finite coupling structure through the many properties derived from crossing and used to bootstrap(19).…”