Causality and the threshold anomaly of the nucleus-nucleus potential

According to the causality principle, a scattered wave cannot be emitted before the arrival of the incident wave. This principle implies the existence of a dispersion relation between the real and the imaginary parts of the optical potential. We discuss the difference between the dispersion relations which hold for nucleus-nucleus scattering on the one hand and for nucleon-nucleus scattering on the other hand. In the case of nucleus-nucleus scattering, the dispersion relation predicts that the modulus of the real part of the optical potential has a bell-shaped maximum, as a function of energy, when the imaginary part approaches zero, i.e. for energies near the top of the Coulomb barrier. The shape of this apparent anomaly is investigated in the framework of several models. It is shown that there exists an algebraic model which is at the same time simple and sufficiently accurate in the sense that the difference between its outcome and that of more realistic models is smaller than the uncertainties introduced by the assumptions which have to be made. Various systems are discussed, in particular 16O + 280Pb and α + 40Ca. Several implications of the anomaly are pointed out, including its effect on the sub-barrier fusion of two heavy ions.