Multi-loop amplitudes in maximally supersymmetric pure spinor field theory

TL;DR

This work develops a BRST-consistent, first-quantised pure-spinor world-line formalism for maximally supersymmetric Yang–Mills theory and supergravity, enabling explicit construction of multi-loop amplitudes that preserve manifest supersymmetry. By introducing a non-minimal pure-spinor sector and a composite b ghost, the authors derive amplitude prescriptions, address zero-mode saturation with regulators, and demonstrate the necessity of higher-point contact terms for BRST invariance. They perform a detailed skeleton-based analysis of one- to five-loop four-point amplitudes, showing how UV divergences arise and how the leading interactions F^4, R^4, and their higher-derivative partners are constrained by supersymmetry, including an explicit indication that the ∂^8 R^4 term is not protected and receives contributions from all loops. The results illuminate the structure of planar and non-planar contributions, clarify the open/closed string parallels, and suggest a path toward a second-quantised, purely cubic action for these theories, with potential connections to color–kinematics duality and late-loop divergences in N=8 supergravity.

Abstract

This paper provides a more detailed background to the results of arXiv:1004.2692 concerning properties of multi-loop amplitudes in a pure spinor formulation of field theories with maximal supersymmetry. This involves the development of a first quantised field theory version of the non-minimal pure spinor formalism originally designed for describing superstring amplitudes. In addition to superspace world-line fields, the formalism involves a set of world-line ghost fields that are required for implementing BRST invariance with a composite b ghost. In particular, we show that BRST invariance requires the presence of certain contact terms. For four-point amplitudes, these are important beyond two loops. We also present an alternative proof of the "no-triangle hypothesis" and the vanishing of amplitudes with fewer than four external particles.

Figures (15)

• Figure 1: The scalar vertex $\left<V_B;\tau\right|$ where the arrows on each line indicate the direction of increasing proper time.
• Figure 2: (a) The function describing absorption of one physical state with momentum $k_r$ ($k^2$) of the three-point vertex. (b) The propagator absorbing $N$ physical particles evaluated between two general states with momenta $p_i$ and $p_f$.
• Figure 3: The figures describing the $s$-, $t$- and $u$-channel of the four-point function. Arrows indicate increasing proper time and the numbers indicate the particle. Figure (a), (b) and (c) illustrate the $s$-, $t$- and $u$-channels, respectively.
• Figure 4: (a) A one-particle irreducible skeleton. (b) A one-particle reducible skeleton that consists of two sub-diagrams with loops. (c) A one-particle irreducible skeleton with a three-point tree sub-diagram attached. Only (a) and (b) will be considered in this paper.
• Figure 5: The unique one-particle irreducible two-loop skeleton diagram. The amplitude is obtained by attaching vertex operators to points on the lines, which are integrated around the diagram. The circular arrows denote the different $B_I$-cycles. The propagators in the skeleton are numbered from 1 to 3 and the arrows on each line indicate the direction of increasing proper time along the line.