Tomasz Lukowski

Large spin systematics in CFT

A bstractUsing conformal field theory (CFT) arguments we derive an infinite number of constraints on the large spin expansion of the anomalous dimensions and structure constants of higher spin operators. These arguments rely only on analyticity, unitarity, crossing-symmetry and the structure of the conformal partial wave expansion. We obtain results for both, perturbative CFT to all order in the perturbation parameter, as well as non-perturbatively. For the case of conformal gauge theories this provides a proof of the reciprocity principle to all orders in perturbation theory and provides a new “reciprocity” principle for structure constants. We argue that these results extend also to non-conformal theories.

Twist-three at five loops, Bethe Ansatz and wrapping

We present a formula for the five-loop anomalous dimension of N = 4 SYM twist-three operators in the sl(2) sector. We obtain its asymptotic part from the Bethe Ansatz and finite volume corrections from the generalized Luscher formalism, considering scattering processes of spin chain magnons with virtual particles that travel along the cylinder. The complete result respects the expected large spin scaling properties and passes non-trivial tests including reciprocity constraints. We analyze the pole structure and find agreement with a conjectured resummation formula. In analogy with the twist-two anomalous di- mension at four-loops wrapping effects are of order (log 2 M/M 2 ) for large values of the spin.

Lessons from crossing symmetry at large N

A bstractWe consider the four-point correlator of the stress tensor multiplet in N=4

N = 4 Scattering Amplitudes and the Deformed Graßmannian

mathcal{N}=4

Yangian symmetry for the tree amplituhedron

SYM. We construct all solutions consistent with crossing symmetry in the limit of large central charge c ∼ N2 and large g2N . While we find an infinite tower of solutions, we argue most of them are suppressed by an extra scale Δgap and are consistent with the upper bounds for the scaling dimension of unprotected operators observed in the numerical superconformal bootstrap at large central charge. These solutions organize as a double expansion in 1/c and 1/Δgap. Our solutions are valid to leading order in 1/c and to all orders in 1/Δgap and reproduce, in particular, instanton corrections previously found. Furthermore, we find a connection between such upper bounds and positivity constraints arising from causality in flat space. Finally, we show that certain relations derived from causality constraints for scattering in AdS follow from crossing symmetry.

Perturbative Four-Point Functions from the Analytic Conformal Bootstrap

Planar N = 4 supersymmetric Yang-Mills theory appears to be integrable. While this allows one to find this theorys exact spectrum, integrability has hitherto been of no direct use for scattering amplitudes. To remedy this, we deform all scattering amplitudes by a spectral parameter. The deformed tree-level four-point function turns out to be essentially the one-loop R matrix of the integrable N = 4 spin chain satisfying the Yang-Baxter equation. Deformed on-shell three-point functions yield novel three-leg R matrices satisfying bootstrap equations. Finally, we supply initial evidence that the spectral parameter might find its use as a novel symmetry-respecting regulator replacing dimensional regularization. Its physical meaning is a local deformation of particle helicity, a fact which might be useful for a much larger class of nonintegrable four-dimensional field theories.

Tree-level scattering amplitudes from the amplituhedron

Some time ago the general tree-level scattering amplitudes of N = 4 Super Yang-Mills theory were expressed as certain Grasmannian contour integrals. These remarkable formulas allow to clearly expose the super-conformal, dual super-conformal, and Yangian symmetries of the amplitudes. Using ideas from integrability it was recently shown that the building blocks of the amplitudes permit a natural multi-parameter deformation. However, this approach had been criticized by the observation that it seemed impossible to reassemble the building blocks into Yangian-invariant deformed non-MHV amplitudes. In this note we demonstrate that the deformations may be succinctly summarized by a simple modification of the measure of the Grasmannian integrals, leading to a Yangianinvariant deformation of the general tree-level amplitudes. Interestingly, the deformed building-blocks appear as residues of poles in the spectral parameter planes. Given that the contour integrals also contain information on the amplitudes at loop-level, we expect the deformations to be useful there as well. In particular, applying meromorphicity arguments, they may be expected to regulate all notorious infrared divergences. We also point out relations to Gelfand hypergeometric functions and the quantum KnizhnikZamolodchikov equations.

Expanding the Bethe/Gauge dictionary

A bstractIt has been recently conjectured that scattering amplitudes in planar N=4