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

Abstract

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. This may be of interest to some dispersive tests of the Standard Model with hadrons, and to unitarization methods used to extend electroweak effective theories. Interestingly, the Inverse Amplitude Method is safe (as the inverse amplitude has the opposite, convergent behavior allowing contour closure). Generically, one-dimensional sum rules such as for the photon vacuum polarization, one-dimensional form factors or the Adler function, to name a few, are not affected by this uncertainty. Likewise, fixed-t dispersion relations were cleverly constructed to avoid it and consequences therefrom are solid.