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Abstract
We consider a class of planar tree-level four-point functions in mathcal{N} = 4 SYM in a special kinematic regime: one BMN operator with two scalar excitations and three half-BPS operators are put onto a line in configuration space; additionally, for the half-BPS operators a co-moving frame is chosen in flavour space. In configuration space, the four-punctured sphere is naturally triangulated by tree-level planar diagrams. We demonstrate on a number of examples that each tile can be associated with a modified hexagon form-factor in such a way as to efficiently reproduce the tree-level four-point function. Our tessellation is not of the OPE type, fostering the hope of finding an independent, integrability-based approach to the computation of planar four-point functions.
Highlights
We will restrict to a special configuration
We consider a class of planar tree-level four-point functions in N = 4 SYM in a special kinematic regime: one BMN operator with two scalar excitations and three half-BPS operators are put onto a line in configuration space; for the half-BPS operators a co-moving frame is chosen in flavour space
The four-punctured sphere is naturally triangulated by tree-level planar diagrams
Summary
These scalar operators are half-BPS and highest-weight states of the su(4) R-symmetry algebra; their dimension does not receive quantum corrections. L. Already at the first order of perturbation theory, these acquire a non-trivial anomalous dimension encoded in a mixing matrix. It is well known that this mixing problem is equivalent to the diagonalization of the Heisenberg spin-chain Hamiltonian [10]. Cyclicity imposes p1 + p2 = u1 + u2 = 0, yielding that physical states are determined by the quantization condition u. The one-loop anomalous dimension is the spin-chain energy level. We collect the first few eigenstates of the one-loop dilatation operator, along with their anomalous dimension and rapidity u.3 We collect the first few eigenstates of the one-loop dilatation operator in table 1, along with their anomalous dimension and rapidity u.3
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