Scattering of Plane Waves in Self-Dual Yang-Mills Theory

TL;DR

This paper tackles the classical self-dual Yang–Mills equation in $3+1$ dimensions and constructs an exact solution describing the scattering of $n$ plane waves. The authors develop a perturbative expansion in the gauge coupling $g$ and introduce a scattering operator $\\hat{T}$ acting on the universal enveloping algebra, an $n$-vector space, and functions on $[0,1]$, yielding a compact closed-form expression $\\Phi(x)=-i \\vec{\\phi}_0 \\cdot \\frac{1}{1-\\hat{T}} \\vec{\\phi}$. They derive recursion relations for higher-order terms, obtain an explicit minimal solution for the coefficient functions $D(j_1,\\ldots,j_m)$, and assemble the full solution as a sum over iterations of the kernel $T$ with a regularization $P$ to control divergences. The work connects SDYM scattering to integrable-system techniques via an auxiliary linear space, providing a compact operator framework with potential implications for nonlinear gauge dynamics.

Abstract

We consider the classical self-dual Yang-Mills equation in 3+1-dimensional Minkowski space. We have found an exact solution, which describes scattering of $n$ plane waves. In order to write the solution in a compact form, it is convenient to introduce a scattering operator $\hat{T}$. It acts in the direct product of three linear spaces: 1) universal enveloping of $su(N)$ Lie algebra, 2) $n$-dimensional vector space and 3) space of functions defined on the unit interval.