  Heavy-fermion superconductors are prime candidates for novel electron-pairing
states due to the spin-orbital coupled degrees of freedom and electron
correlations. Superconductivity in CeCu$_2$Si$_2$ discovered in 1979, which is
a prototype of unconventional (non-BCS) superconductors in strongly correlated
electron systems, still remains unsolved. Here we provide the first report of
superconductivity based on the advanced first-principles theoretical approach.
We find that the promising candidate is an $s_\pm$-wave state with loop-shaped
nodes on the Fermi surface, different from the widely expected line-nodal
$d$-wave state. The dominant pairing glue is magnetic but high-rank octupolar
fluctuations. This system shares the importance of multi-orbital degrees of
freedom with the iron-based superconductors. Our findings reveal not only the
long-standing puzzle in this material, but also urge us to reconsider the
pairing states and mechanisms in all heavy-fermion superconductors.
