Space-time S-matrix and Flux-tube S-matrix III. The two-particle contributions

Abstract

We consider light-like Wilson loops with hexagonal geometry in the planar limit of N=4 Super-Yang-Mills theory. Within the Operator-Product-Expansion framework these loops receive contributions from all states that can propagate on top of the colour flux tube sourced by any two opposite edges of the loops. Of particular interest are the two-particle contributions. They comprise virtual effects like the propagation of a pair of scalars, fermions, and gluons, on top of the flux tube. Each one of them is thoroughly discussed in this paper. Our main result is the prediction of all the twist-2 corrections to the expansion of the dual 6-gluons MHV amplitude in the near-collinear limit at finite coupling. At weak coupling, our result was recently used by Dixon, Drummond, Duhr and Pennington to predict the full amplitude at four loops. At strong coupling, it allows us to make contact with the classical string description and to recover the (previously elusive) AdS(3) mode from the continuum of two-fermion states. More generally, the two-particle contributions serve as an exemplar for all the multi-particle corrections.

Figures (24)

• Figure 1: In the near collinear limit $\tau \to \infty$, the Wilson loop $\mathcal{W}$ has an expansion in the number of particles flowing in the color flux tube. Here we depict the three leading contributions corresponding to the vacuum, single-particle and two-particle states, respectively.
• Figure 2: Masses $m(g)$ of the twist-one excitations as functions of the coupling $g = \sqrt{\lambda}/(4\pi)$. The lightest excitations are the scalar ones. Their mass defines the mass gap of the theory and it becomes exponentially small at strong coupling AldayMaldacena. Its plot above agrees with the one in Fioravanti:2008rv which studied the related problem of solving the Freyhult-Rej-Staudacher equation Freyhult:2007pz. The mass of the fermions is protected by supersymmetry AldayMaldacena while the one of the gluons interpolates between $1$ and $\sqrt{2}$.
• Figure 3: Table of fundamental excitations, represented by the squares in this figure. The twist-one excitations sit on the diagonal and their multiplicities follow from the dimensions of their $SU(4)$ representations. The two towers of gluon bound states, with twist $2, 3, \ldots$, and $U(1)$ charge $\pm 2, \pm 3, \ldots$, are depicted on the two semi-infinite lines at the top and bottom of the table. Multi-particle states made out of these excitations span the flux-tube Hilbert space at any coupling.
• Figure 4: a) The single-gluon contribution ${\cal W}_F$ at $g=1/2$. b) The two-gluon contributions ${\cal W}_{FF}$ (bottom/red), ${\cal W}_{DF}$ (middle/blue) and ${\cal W}_{F\bar{F}}$ (top/yellow) at $g=1/2$. Contributions ( b) comprising heavier excitations are seen to be smaller than the single-particle one ( a), as expected (for positive $\tau$). As we increase $\tau$ or $\sigma$, the excitations have to propagate over larger distance and their contributions get more suppressed, as shown in both a) and b).
• Figure 5: The fermion integration contour covers all real (non-zero) momenta which in terms of the Bethe rapidity $u$ amounts to a contour in a two-sheeted Riemann surface. The wavy line stretching between $u=\pm 2g$ stands for the cut connecting the two $u$-planes.