  Background: The Density-constraint Time-dependent Hartree-Fock method is
currently the tool of choice to predict fusion cross-sections. However, it does
not include pairing correlations, which have been found recently to play an
important role. Purpose: To describe the fusion cross-section with a method
that includes the superfluidity and to understand the impact of pairing on both
the fusion barrier and cross-section. Method: The density-constraint method is
tested first on the following reactions without pairing, $^{16}$O+$^{16}$O and
$^{40}$Ca+$^{40}$Ca. A new method is developed, the Density-constraint
Time-dependent Hartree-Fock-Bogoliubov method. Using the Gogny-TDHFB code, it
is applied to the reactions $^{20}$O+$^{20}$O and $^{44}$Ca+$^{44}$Ca. Results:
The Gogny approach for systems without pairing reproduces the experimental data
well. The DC-TDHFB method is coherent with the TDHFB fusion threshold. The
effect of the phase-lock mechanism is shown for those reactions. Conclusions:
The DC-TDHFB method is a useful new tool to determine the fusion potential
between superfluid systems and to deduce their fusion cross-sections.
