M5-Brane and D-Brane Scattering Amplitudes

TL;DR

This work develops explicit tree-level, on-shell scattering amplitudes for three brane theories with 16 supercharges: D3 in 4D, and D5 and M5 in 6D, using twistor-string–like formulas. The D3 amplitudes exhibit helicity conservation due to an SU(4)×U(1) R-symmetry, while the 6D D5 and M5 amplitudes are shown to reduce to the D3 results under dimensional reduction, establishing a unifying framework. The authors propose a general M5 formula A_n = ∫ d^nσ dM Δ_B Δ_F det′ S_n U(ρ,σ) that reproduces D3 amplitudes upon reduction, with U factorized to ensure correct 6D invariances and factorization properties; they further derive and verify six- and eight-particle M5 amplitudes, discuss SL(2,C) modular symmetry, soft theorems, and the role of a generalized resultant R(ρ). They also present analogous results for the D5 theory, highlighting deep connections among the three theories and offering insights toward a possible 6D twistor-string origin of the M5 S-matrix. The work provides concrete, symmetry-rich S-matrix constructions for brane theories and sets the stage for further exploration of their geometric and string-theoretic underpinnings.

Abstract

We present tree-level $n$-particle on-shell scattering amplitudes of various brane theories with $16$ conserved supercharges. These include the world-volume theory of a probe D3-brane or D5-brane in 10D Minkowski spacetime as well as a probe M5-brane in 11D Minkowski spacetime, which describes self interactions of an abelian tensor supermultiplet with 6D $(2,0)$ supersymmetry. Twistor-string-like formulas are proposed for tree-level scattering amplitudes of all multiplicities for each of these theories. The R symmetry of the D3-brane theory is shown to be $SU(4) \times U(1)$, and the $U(1)$ factor implies that its amplitudes are helicity conserving. Each of 6D theories (D5-brane and M5-brane) reduces to the D3-brane theory by dimensional reduction. As special cases of the general M5-brane amplitudes, we present compact formulas for examples involving only the self-dual $B$ field with $n=4,6,8$.

Figures (2)

• Figure 1: Exchange diagrams contributing to the 6 $B_{++}$ amplitude. The internal line may be any of the three states, and we sum over all the factorization channels as well. It is important to note that these diagrams do not come directly from Feynman rules as there is no covariant action available for the M5 theory; instead, they represent the factorization of the amplitude at the poles where $s_{ijk} \rightarrow 0$.
• Figure 2: Diagrammatic expression of the local term for a six-particle amplitude. In the example where all external particles are $B_{ab}$, this local term vanishes, and the exchange diagrams are the only contribution to the total amplitude.