Smearing of causality by compositeness divides dispersive approaches into exact ones and precision-limited ones

Understand

Scattering off the edge of a composite particle or finite-range interaction can precede that off its center.

• An effective theory treatment with pointlike particles and contact interactions must find that the scattered experimental wave is slightly advanced, in violation of causality (the fundamental underlying theory being causal).
• In practice, partial-wave or other projections of multivariate amplitudes exponentially grow with imaginary E, so that upper complex-plane analyticity is not sufficient to obtain a dispersion relation for them, but only for a slightly modified function (the modified relations additionally connect different J).
• This limits the maximum precision of certain dispersive approaches to compositeness based on Cauchy's theorem leading to partial-wave dispersion relations.