Gamow-Teller transition strengths in the intermediate nucleus of the $^{116}$Cd double-$\beta$ decay by the $^{116}$Cd$(p,n)$ and $^{116}$Sn$(n,p)$ reactions at 300~MeV

Gamow-Teller (GT) transition strengths in the intermediate nucleus of the $^{116}$Cd double-$\beta$ ($\beta\beta$) decay, namely $^{116}$In have been studied by measuring the double differential cross sections for the $^{116}$Cd$(p,n)$ and $^{116}$Sn$(n,p)$ reactions at 300~MeV over a wide excitation-energy region including GT giant resonance (GTGR). A large amount of the strengths in the $\beta^+$ direction has been newly found in the GTGR region ($E_x=5$ to 20~MeV), which may indicate that a large part of the nuclear matrix element of the two-neutrino $\beta\beta$ decay comes from this region as well as the cancellation due to phase.