Gamow-Teller strengths in $^{48}$Ca with the charge-exchange subtracted second random-phase approximation

We develop a fully self-consistent subtracted second random-phase approximation for charge-exchange processes with Skyrme energy-density functionals. As a first application, we study Gamow-Teller excitations in the doubly-magic nucleus $^{48}$Ca, which is the lightest double-$\beta$ emitter that could be used in an experiment. The amount of Gamow-Teller strength below 20 or 30 MeV is considerably smaller than in other energy-density-functional calculations and agrees better with experiment. This important result, obtained without \textit{ad hoc} quenching factors, is due to the presence of two-particle -- two-hole configurations. Their density progressively increases with excitation energy, leading to a long high-energy tail in the spectrum, a fact that may have implications for the computation of nuclear matrix elements for neutrinoless double-$\beta$ decay in the same framework.