Dual conformal symmetry at loop level; massive regularization

TL;DR

This review synthesizes the status of dual conformal symmetry in planar N=4 SYM at loop level, arguing that infrared divergences obscure the symmetry unless an AdS/CFT-inspired regulator is used. By employing the Coulomb branch with a mass regulator, the author shows an exact extended dual conformal symmetry that constrains the loop integrand and reduces the loop-integral basis to a small, well-behaved set of invariants. The work discusses practical construction of loop amplitudes via unitarity and recursion, advances in evaluating the resulting integrals with Mellin-Barnes methods and momentum twistors, and highlights differential equations that relate higher-loop integrals to lower-loop ones. The overall aim is to bridge weak and strong coupling descriptions and to realize all-loop results through a tightly constrained, symmetry-guided framework.

Abstract

Dual conformal symmetry has had a huge impact on our understanding of planar scattering amplitudes in N=4 super Yang-Mills. At tree level, it combines with the original conformal symmetry generators to a Yangian algebra, a hallmark of integrability, and helps in determining the tree-level amplitudes. The latter are now known in closed form. At loop level, it determines the functional form of the four- and five-point scattering amplitudes to all orders in the coupling constant, and gives restrictions at six points and beyond. The symmetry is best understood at loop level in terms of a novel AdS-inspired infrared regularization which makes the symmetry exact, despite the infrared divergences. This has important consequences for the basis of loop integrals in this theory. Recently, a number of selective reviews have appeared which discuss dual conformal symmetry, mostly at tree level. Here, we give an up-to-date account of dual conformal symmetry, focussing on its status at loop level.

Figures (5)

• Figure 1: Loop integrals appearing up to three loops in the four-point amplitude. Numerator factors independent of the loop momentum and a loop-dependent numerator in diagram (d) are not displayed.
• Figure 2: Dual representation of the integrals of Fig. \ref{['fig-loop']}. Vertices denote dual integration points. The dashed line in Fig. (d) denotes an internal numerator factor. The latter is required by dual conformal symmetry, as explained in the text.
• Figure 3: (a) String theory description for the scattering of $M$ gluons in the large $N$ limit. Putting the $M$ D3-branes at different positions $z_{i}\neq 0$ serves as a regulator and also allows to exhibit dual conformal symmetry. (b) Gauge theory analogue of (a): a sample two-loop integral at large $N$, in double line notation. Mixed full/dashed lines correspond to massive propagators. Picture from Alday:2009zm.
• Figure 4: Factorization of the leading-log and next-to-leading-log contributions to the Regge limit $s \gg t$ of the $L$-loop vertical ladder integral $I_{L\,a}(v,u)$ into simpler integrals. Factorization of the NLL contribution of the vertical ladder integral with H-shaped insertion $I_{L\,H}$Henn:2010bkHenn:2010ir. The dotted line indicates a loop-momentum-dependent numerator. Picture from Henn:2010ir.
• Figure 5: The integral of equation (\ref{['eq-yangian-int']}) in momentum space (a) and in dual notation (b). The position space variables $y_{i}^{\mu}$ are related to the momenta $p^{\mu}_{i}$ by Fourier transform, the dual coordinates $x^{\mu}$ are defined by equation (\ref{['dualcoordinates']}). The original and the dual diagram are both built from quartic vertices only, so that the integral has both a conformal as well as a dual conformal symmetry.