  We study Coulomb correlation effects and their role in superconductivity of
$\varepsilon$-iron under pressure from 12 to 33 GPa by using a combination of
density functional and dynamical mean-field theory. Our results indicate a
persistence of the Fermi-liquid behavior below the temperature $\sim$1000 K.
The Coulomb correlations are found to substantially renormalize the density of
states, reducing the distance from the peak to the Fermi level to 0.4 eV
compared to 0.75 eV obtained in DFT calculations. We find significant
antiferromagnetic correlations, which are accompanied by the formation of
short-lived local magnetic moments. We use the obtained results as a starting
point for construction of the multi-band Bethe-Salpeter equation, which
eigenvalues indicate that antiferromagnetic spin fluctuations may result in the
superconducting pairing in $\varepsilon$-Fe. Moreover, the tendency to
superconducting instability becomes weaker with the increase of pressure, which
may explain the disappearance of superconductivity at $\sim$30 GPa.
