Transverse tails and higher order moments

The tails that may be engendered in a beam´s transverse phase space distribution by, e.g., intrabunch wakefields and nonlinear magnetic fields, are an important diagnostic and object of tuning in linear colliders. Wire scanners or phosphorescent screen monitors yield one dimensional projected spatial profiles of such beams that are generically asymmetric around their centroids, and therefore require characterization by the third moment ⟨x³⟩ in addition to the conventional mean-square or second moment. A set of measurements spread over sufficient phase advance then allows the complete set ⟨x³⟩, ⟨xx´²⟩, ⟨x´³⟩, and ⟨x²x´⟩ to be deduced-the natural extension of the well-known `emittance measurement´ treatment of second moments. The four third moments may be usefully decomposed into parts rotating in phase space at the β-tron frequency and at its third harmonic, each specified by a phase-advance-invariant amplitude and a phase. They provide a framework for the analysis and tuning of transverse wakefield tails