Space-time S-matrix and Flux-tube S-matrix IV. Gluons and Fusion

TL;DR

The paper develops a comprehensive bootstrap for the flux-tube gluonic sector in planar N=4 SYM, showing that multi-gluon pentagon transitions factorize into two-particle building blocks and can be extended to bound states via fusion. It introduces charged (NMHV) transitions and demonstrates that NMHV data can be obtained by multiplying MHV transitions with simple form factors, with h_a(u) determined through half-mirror fusion. The authors construct bound-state transitions and S-matrices from constituent gluons, verify square-limit consistency, and apply the formalism to compute MHV and NMHV hexagons (and the NMHV heptagon) at finite coupling, obtaining all-loop expressions and strong agreement with perturbative data up to four loops. These results substantiate the integrability-based pentagon/OPE framework as a powerful, cross-checkable tool for gluon scattering amplitudes and suggest a viable path toward a full, all-loop bootstrap of higher-point amplitudes in this theory.

Abstract

We analyze the pentagon transitions involving arbitrarily many flux-tube gluonic excitations and bound states thereof in planar N=4 Super-Yang-Mills theory. We derive all-loop expressions for all these transitions by factorization and fusion of the elementary transitions for the lightest gluonic excitations conjectured in a previous paper. We apply the proposals so obtained to the computation of MHV and NMHV scattering amplitudes at any loop order and find perfect agreement with available perturbative data up to four loops.

Figures (8)

• Figure 1: Picture of flux tube excitations and their quantum numbers. Lying on the diagonal are the twist-one excitations which can be scalar $\phi$, fermionic $\psi, \bar{\psi}$, or gluonic $F, \bar{F}$. The latter excitations can form bound states depicted on the bottom and top rows. There is precisely one bound state at any given twist $2, 3, \ldots$ and $U(1)$ charge $\pm 2, \pm 3, \ldots\,$, denoted by $DF\sim F^2, D^2F\sim F^3, \ldots$ or their complex conjugates. In this paper we consider the OPE contribution from states made out of any number of gluons and bound states, that is built out of the excitations presented in the boldfaced squares only.
• Figure 2: (a) The inverse mirror transformation $u\rightarrow u^{-\gamma}$ sends an excitation to the neighbouring edge on the right and simultaneously flips its $U(1)$ charge. (b) A sequence of five mirror rotations sends the excitation all the way around the pentagon.
• Figure 3: In the notations of data, the hexagon NMHV component ${\cal W}^{(1111)}$ corresponds to charging the bottom edge (here numbered 1) with four $\eta$'s. At tree level, this component contains the insertion of a gauge field ${\color{blue}F\color{black}}$ at the bottom cusp superloopskinnersuperloopsimon. In the OPE decomposition, charging the bottom edge this way amounts to replacing the bottom creation pentagon transition $P(0|\psi)$ with the charged transition $P^*(0|\psi)$.
• Figure 4: The four possible charged transitions for the creation or annihilation of a single gluon excitation, ${\color{blue}F\color{black}}$ or ${\color{red}\bar{F}\color{black}}$. The equalities in the figure follow from the rotation symmetry of the pentagon.
• Figure 5: A bound state of $n$ gluons can be described by a vertical string of $n$ Bethe rapidities separated by $i$. The right prescription is to build the string for a centre-of-mass rapidity $u$ lying within the strip $-2g<\text{Re}\,(u)<2g$. This (typically) means that the string is sitting in-between the branch points present in the complex rapidity plane of a (twist-one) gluon. This is what is shown here with the crosses representing the branch points at $\pm 2g \pm i/2$ and the dashed lines the outward cuts connecting them.