  Background: Spin-triplet ($S=1$) proton-neutron (pn) pairing in nuclei has
been under debate. It is well known that the dynamical pairing affects the
nuclear matrix element of the Gamow-Teller (GT) transition and the double beta
decay. Purpose: We investigate the effect of the pn-pair interaction in the
$T=0, S=1$ channel on the low-lying spin-dipole (SD) transition. We then aim at
clarifying the distinction of the role in between the SD and GT transitions.
Method: We perform a three-body model calculation for the transition
${}^{80}\mathrm{Ni}\to{}^{80}\mathrm{Cu}$, where ${}^{78}\mathrm{Ni}$ is taken
as a core. The strength of the pair interaction is varied to see the effect on
the SD transition-strength distribution. To fortify the finding obtained by the
three-body model, we employ the nuclear energy-density functional method for
the SD transitions in several nuclei, where one can expect a strong effect.
Results: The effect of the $S=1$ pn-pair interaction depends on the spatial
overlap of the pn pair and the angular momentum of the valence nucleons; the
higher the angular momentum of the orbitals, the more significant the effect.
Conclusions: The dynamical $S=1$ pairing is effective even for SD states
although the spatial overlap of the pn pair can be smaller than GT states. The
SD transition involving high-$\ell$ orbitals with the same principal quantum
number is strongly affected by the dynamical $S=1$ pairing.
