Recently, with new Time-dependent Hartree-Fock-Bogoliubov codes, the effect of the relative gauge angle in reactions has been studied. It has been shown that the fusion barrier is changed when the initial relative gauge angle between the two fragments is changed. This effect is expected to play an important role in fusion reactions since it is predicted that it can induce fluctuation of the fusion barrier up to 15 MeV. Nevertheless, in nuclear reactions, the number of particles is small and initially fixed. Then to understand the results predicted by TDHFB, it is necessary to restore the initial particle-number symmetry. The effect of the gauge angle will be studied in a simple toy model, where it can be possible to compare the results of TDHFB with the exact solution.
On the second part of the presentation, I will examine the empirical evidence that shows that the fusion barrier width is enhanced in reactions where both fragments are superfluid. For that, two new methods will be presented. First, a method to compute the fusion barrier distribution using the local regression method. Second, a direct calculation of the fluctuations of the barrier distribution that does not need the calculation of the second derivative of the fusion cross section. This second method that shows the best performance has been applied to 115 systems where experimental fusion cross-sections have been measured.