Exploring straight infinite Wilson lines in the self-dual and the MHV Lagrangians

We investigate the appearance of straight infinite Wilson lines lying on the self-dual plane in the context of the self-dual sector of the Yang-Mills theory and in a connection to the Lagrangian implementing the maximally helicity violating (MHV) vertices (MHV Lagrangian) according to the Cachazo-Svrcek-Witten method. It was already recognized in the past by two of the authors, that such Wilson line functional provides the field transformation of positive helicity fields between the Yang-Mills theory on the light cone and the MHV Lagrangian. Here we discuss in detail the connection to the self-dual sector and we provide a new insight into the solution for the minus helicity field transformation, which can be expressed in terms of a functional derivative of the straight infinite Wilson line on the self-dual plane.