Gluon scattering on the self-dual dyon

The computation of scattering amplitudes in the presence of non-trivial background gauge fields is an important but extremely difficult problem in quantum field theory. In even the simplest backgrounds, obtaining explicit formulae for processes involving more than a few external particles is often intractable. Recently, it has been shown that remarkable progress can be made by considering background fields which are chiral in nature. In this paper, we obtain a compact expression for the tree-level, maximal helicity violating (MHV) scattering amplitude of an arbitrary number of gluons in the background of a self-dual dyon. This is a Cartan-valued, complex gauge field sourced by a point particle with equal electric and magnetic charges and can be viewed as the self-dual version of a Coulomb field. Twistor theory enables us to manifest the underlying integrability of the self-dual dyon, trivializing the perturbative expansion in the MHV sector. The formula contains a single position-space integral over a spatial slice, which can be evaluated explicitly in simple cases. As an application of the formula, we show that the holomorphic collinear splitting functions of gluons in the self-dual dyon background are un-deformed from a trivial background, meaning that holomorphic celestial OPE coefficients and the associated chiral algebra are similarly un-deformed. We also comment on extensions of our MHV formula to the full tree-level gluon S-matrix.