  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.
