  In a previous paper (hep-th/0510080) the effective Landau-Lifshitz (LL)
Lagrangians in the SU(2) sector coming from string theory and gauge theory have
been found to three loops in the effective expansion parameter
$\tilde\lambda=\lambda/J^2$. In this paper we continue this study and find the
effective Landau-Lifshitz Lagrangians to four loops. We extend to four-loops
$\tilde\lambda^4$ the computations of 1/J and $1/J^2$ corrections to BMN
energies done in hep-th/0510080 to three-loops. We compare these corrections
obtained from quantum ``gauge-theory'' LL action with the corrections obtained
from the conjectured Bethe ansatz for the long range spin chain representing
perturbative {large $N$} $\NN=4$ Super Yang-Mills in the SU(2) sector and find
perfect matching to four loops $\tilde\lambda^4$. We compare also the 1/J and
$1/J^2$ corrections obtained from quantum ``string-theory'' LL action with
those obtained from the "quantum string" Bethe ansatz and again find perfect
matching to four-loops.
