Scattering of very light charged particles.

I advance arguments against the view that the Lee-Nauenberg-Kinoshita theorem is relevant in practice to the scattering of charged particles as their mass tends to zero. I also discuss the case of massive coloured particle scattering. Scattering cross-sections for reactions involving, in the initial state, a charged particle of mass m contain (in perturbation theory) ln m terms, which become large as m → 0. These logarithms come from virtual photons which are nearly parallel to the charged particle. In 1964, following earlier work of Kinoshita [1], Lee and Nauenberg [1] proved that by summing over an appropriate set of initial states the ln m terms could be removed. Since then it seems not to have been generally agreed whether the initial states required for the theorem correspond to physically realistic situations. (For further references, see [2].) In 1992, Contopanagos and Einhorn [2] (referred to as CE below) published a paper which, amongst many other things, seemed to claim with some certainty that the Lee-Nauenberg-Kinoshita (KLN) initial-state sum does represent physical reality. They went further and studied qualitatively the extent of the initial-state sum necessary to represent a typical physical, realistic situation. This has the great virtue of focusing the argument in a concrete way. I remain unconvinced of the physical relevance of the initial-state sums, and, lest the very thorough CE paper should be thought to close the argument, I write this note to emphasize the questions which, to my mind, remain. I do not claim any complete understanding, but I hope this note may at any rate provoke further discussion. In QED, soft divergences are well understood in the Bloch-Nordsiek theory. Further, the application of the KLN theorem to final-state collinear divergences is uncontroversial. So I concentrate on initial-state collinear divergences in QED. I also discuss the relevance of the KLN theorem to soft divergences in (perturbative) QCD, where, even with massive particles in the initial state, there are infrared divergences (uncancelled by Bloch-Nordsiek final-state sums) [3]. First note that the opinion of CE is not totally clear. In the second paragraph of the paper they state 'we shall show that the requisite initial-state sum does inevitably occur in massless theories'. But in Section 4 they say: 'The equality displayed in Eq. (4.1) requires a specific relative weighting among degenerate initial states, viz., the same 1