Scattering into the fifth dimension of N=4 super Yang-Mills

TL;DR

The paper introduces a Higgsed version of N=4 SYM as a four-dimensional infrared regulator for planar amplitudes, motivated by AdS/CFT. In this framework, masses from spontaneous symmetry breaking extend the dual conformal symmetry to an effective five-dimensional space, drastically restricting the allowed loop integrals and eliminating triangle subgraphs. The authors perform explicit tests: a one-loop four-point scalar amplitude exhibits exact dual conformal invariance in the five-dimensional embedding, and two-loop results demonstrate exponentiation of the finite part with a cusp anomalous dimension matching the known value. They argue the finite parts are regulator-independent and consistent with anomalous Ward identities, and outline extensions to higher points and the relation to dual superconformal symmetry, with several promising directions for future work.

Abstract

We study an alternative to dimensional regularisation of planar scattering amplitudes in N=4 super Yang-Mills theory by going to the Coulomb phase of the theory. The infrared divergences are regulated by masses obtained from a Higgs mechanism, allowing us to work in four dimensions. The corresponding string theory set-up suggests that the amplitudes have an exact dual conformal symmetry. The latter acts on the kinematical variables of the amplitudes as well as on the Higgs masses in an effectively five dimensional space. We confirm this expectation by an explicit calculation in the gauge theory. A consequence of this exact dual conformal symmetry is a significantly reduced set of scalar basis integrals that are allowed to appear in an amplitude. For example, triangle sub-graphs are ruled out. We argue that the study of exponentiation of amplitudes is simpler in the Higgsed theory because evanescent terms in the mass regulator can be consistently dropped. We illustrate this by showing the exponentiation of a four-point amplitude to two loops. Finally, we also analytically compute the small mass expansion of a two-loop master integral with an internal mass.

Figures (7)

• Figure 1: (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 us to exhibit dual conformal symmetry. (b) Gauge theory analogue of (a): a generic scattering amplitude at large $N$ (here: a sample two-loop diagram).
• Figure 2: Original (left) and dual (right) pictures of a scattering amplitude. On the original picture the open strings end at $D3-$branes located at $z_i=m_i$. In the dual picture we have open strings stretched between $D-$instantons separated by a light-like distance.
• Figure 3: (a) Double line notation of the gauge factor corresponding to a one-loop box integral. The $U(M)$ indices $i_{n}$ determine the masses of the different propagators. (b) Dual diagram (thick black lines) and dual coordinates. The fifth component of the dual coordinates corresponds to the radial AdS${}_{5}$ direction.
• Figure 4: (a) Double line notation of the gauge factor corresponding to the two-loop box integral in the Higgsed theory. The integral is dual conformally invariant. (b) Diagram for the same integral in the equal mass case $m_{i}=m$. Dashed thin lines denote massless propagators, thick black lines denote massive propagators.
• Figure 5: (a) An example of a higher-point dual conformal integral. The picture corresponds to a '1-mass' integral, since the sum $p^{\mu}_{3}+p^{\mu}_{4}$ is in general not light-like. As in the four-point case, there are the masses of the Higgsed particles circulating in the outer loops. (b) In the equal mass case $m_{i}=m$, all outer legs become massless (dashed lines), while the internal propagators (full black lines) have uniform mass $m$, making the integral infrared finite.