Scattering Amplitudes -- Wilson Loops Duality for the First Non-planar Correction

TL;DR

This work extends the planar amplitude/Wilson-loop duality of ${\cal N}=4$ SYM to the first non-planar correction, the leading 1/$N$ double-trace amplitude $A_{n,m}$, by introducing a cylinder (two infinite null Wilson lines) dual with a quantum periodicity constraint. The cylindrically cut amplitude ${\mathbb A}_{n,m}(l)$ is made physical by summing over windings $l \to l + a q$ and is dual to a cylinder Wilson lines correlator ${\widehat{{\cal W}}_{n,m}(l,\theta)}$, with the master relation ${\mathbb M}_{n,m}(l) = \int d^8\theta\,{\widehat{{\cal W}}_{n,m}(l,\theta)}$. The authors substantiate the duality in a solvable fishnet model and show, at one loop in ${\cal N}=4$ SYM, that the cylindrically cut amplitude and the Wilson lines correlator agree; they also discuss the AdS string picture, where T-duality relates the cylinder amplitude to a pair of periodic Wilson lines, with a quantum periodicity constraint enforcing the correspondence. They further argue that the cylinder loop integrand can be constructed via recursion and that the pentagon OPE formalism can be extended to first non-planar order, setting a path toward finite-coupling control of non-planar corrections. The framework opens avenues for higher-order $1/N$ corrections and non-planar integrability in a controlled, gauge-invariant setting.

Abstract

We study the first non-planar correction to gluon scattering amplitudes in ${\cal N}=4$ SYM theory. The correction takes the form of a double trace partial amplitude and is suppressed by one power of $1/N$ with respect to the leading single trace contribution. We extend the duality between planar scattering amplitudes and null polygonal Wilson loops to the double trace amplitude. The new duality relates the amplitude to the correlation function of two infinite null polygonal Wilson lines that are subject to a quantum periodicity constraint. We test the duality perturbatively at one-loop order and demonstrate it for the dual string in AdS. The duality allows us to extend the notion of the loop integrand beyond the planar limit and to determine it using recursion relations. It also allows one to apply the integrability-based pentagon operator product expansion approach to the first non-planar order.

Figures (9)

• Figure 2.1: A Feynman diagram that contributes to the double trace partial amplitude $A_{3,2}$. The dashed lines indicate its various cylinder cuts, which correspond to the diagram with fixed momenta flowing around the cylinder, $l$. The red dashed line wraps around the cylinder once more than the blue one, which implies that the cylinder cut momenta, $l$, is only defined modulo a shift by the momentum flowing through the cylinder, $l\simeq l+q$.
• Figure 2.2: The periodic Wilson lines configuration that is dual to the cylindrically cut double trace amplitude. Each line consists of the ordered gluon momenta in one of the two traces. Because the total momentum in each trace is non-zero, the dual line is not closed. Instead, it is repeated periodically and forms the boundary of the universal cover of the cylinder. The separation between the two Wilson lines is equal to the momentum that flows around the cylinder, $l$.
• Figure 3.1: The leading contribution to the eight-point amplitude of $\{\phi_1(k_1),\phi_1(k_2),\phi_2(k_3),\phi_2(k_4),\phi_1^\dagger(k_5),\phi_1^\dagger(k_6),\phi_2^\dagger(k_7),\phi_2^\dagger(k_8)\}$ in the fishnet model. Here black lines correspond to the propagators in the double line notation and the grey arrows indicates the flow of the two conserved $U(1)$ charges. In the dual space one finds a null octagon Wilson loop, represented by the blue line. It consists of four scalars inserted at $\{x_1,x_3,x_5,x_7\}$. The dashed red lines represent the single Feynman diagram that contributes to the expectation value of the dual octagon. It has a single interaction vertex at $y=l+x_1$.
• Figure 3.2: The two one-loop Feynman diagrams that contribute to the four-point amplitude of $\{\phi_1\left(k_1\right),\phi_1\left(k_2\right),\phi_1^\dagger(k_3),\phi_1^\dagger(k_4)\}$.
• Figure 3.3: The two types of diagrams that contribute to the Wilson lines correlator (\ref{['KazWL']}). The dual space picture consists of two infinite null polygonal Wilson lines composed of two sets of scalars, $\{\phi_2(x_1^{[a]})/(\langle1\,2\rangle c_0^2),\phi_2(x_2^{[a]})/(\langle2\,1\rangle c_0^2)\}$ and $\{\phi^\dagger_2(\dot{x}_1^{[a]})/(\langle3\,4\rangle c_0^2),\phi^\dagger_2(\dot{x}_2^{[a]})/(\langle4\,3\rangle c_0^2)\}$, contracted along the edges. Each propagator between $x_i$ and $\dot x_j$ has infinitely many identical images running between $x_i^{[a]}$ and $\dot x_j^{[a]}$, but is counted only once. Apart from these two diagrams, there are two infinite sets of diagrams that are related to these by shifting one of the lines by an integer number of periods.