An Infinite Set of Tree Amplitudes in Higgs-yang-mills

It is pointed out that the Parke-Taylor or maximally helicity violating amplitudes in the pure Yang-Mills can, after some specifications, be interpreted as amplitudes of scattering of massive vector bosons in the Higgs-Yang-Mills system. Although, in general, multi-particle tree amplitudes are complicated things (because the number of various Feynman diagrams grows very fast with the number of legs), see, e.g., the review [1] and more recent reviews [2], [3], in some instances one can explicitly construct infinite sets of tree amplitudes, as it was done in [4], [5], [6]. The basic example is [4], where nice explicit expressions for the socalled maximally helicity violating (MHV) amplitudes in Yang-Mills (YM) have been conjectured (these expressions have been proved in [7]). MHV amplitudes are gluonic amplitudes with arbitrary number of legs and most of the external gluons in the same helicity states. More precisely, MHV amplitudes describe scattering of two positive helicity gluons into any number of positive helicity gluons. That is the restriction on polarizations which allows one to obtain the explicit formulae for the amplitudes with arbitrary number of legs. Perhaps, the simplest procedure to obtain MHV amplitudes has been developed in [8] following ideas of [9], [10]. The procedure was to obtain the positive helicity gluonic form-factors via a solution of the self-duality equation and the MHV amplitudes via a solution of the linearization of YM equations in the self-dual background. In these terms MHV amplitudes are amplitudes of scattering of two anti-self-dual gluons