  It is a current important subject to clarify properties of chiral
three-nucleon forces (3NFs) not only in nuclear matter but also in scattering
between finite-size nuclei. Particularly for the elastic scattering, this study
has just started and the properties are not understood in a wide range of
incident energies ($E_{\rm in}$). We investigate basic properties of chiral
3NFs in nuclear matter with positive energies by using the
Brueckner-Hartree-Fock method with chiral two-nucleon forces of N$^{3}$LO and
3NFs of NNLO, and analyze effects of chiral 3NFs on $^{4}$He elastic scattering
from targets $^{208}$Pb, $^{58}$Ni and $^{40}$Ca over a wide range of $30
\lesssim E_{\rm in}/A_{\rm P} \lesssim 200$ MeV by using the $g$-matrix folding
model, where $A_{\rm P}$ is the mass number of projectile. In symmetric nuclear
matter with positive energies, chiral 3NFs make the single-particle potential
less attractive and more absorptive. The effects mainly come from the
Fujita-Miyazawa 2$\pi$-exchange 3NF and slightly become larger as $E_{\rm in}$
increases. These effects persist in the optical potentials of $^{4}$He
scattering. As for the differential cross sections of $^{4}$He scattering,
chiral-3NF effects are large in $E_{\rm in}/A_{\rm P} \gtrsim 60$ MeV and
improve the agreement of the theoretical results with the measured ones.
Particularly in $E_{\rm in}/A_{\rm P} \gtrsim 100$ MeV, the folding model
reproduces measured differential cross sections pretty well. Cutoff ($\Lambda$)
dependence is investigated for both nuclear matter and $^{4}$He scattering by
considering two cases of $\Lambda=450$ and $550$ MeV. The uncertainty coming
from the dependence is smaller than chiral-3NF effects even at $E_{\rm
in}/A_{\rm P}=175$ MeV.
