Abstract
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 {lambda}-tilde={lambda}/J{sup 2}. In this paper we continue this study and find the effective Landau-Lifshitz Lagrangians to four loops. We extend to four-loops {lambda}-tilde{sup 4} the computations of 1/J and 1/J{sup 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 N=4 Super Yang-Mills theory in the SU(2) sector and find perfect matching to four loops {lambda}-tilde{sup 4}. We compare also the 1/J and 1/J{sup 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.
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