  For essential reduced hyperplane arrangements of 4 variables, we show that
the pole order spectral sequence degenerates almost at $E_2$, and completely at
$E_3$, generalizing the 3 variable case where the complete $E_2$-degeneration
is known. These degenerations are useful to determine the roots of
Bernstein-Sato polynomials supported at the origin. For the proof we improve an
estimate of the Castelnuovo-Mumford regularity of logarithmic vector fields
which was studied by H. Derksen and J. Sidman.
