Quantum Landau-Lifshitz model at four loops: 1/J and 1/J2 corrections to Berenstein-Maldacena-Nastase energies

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 λ-tilde=λ/J2. In this paper we continue this study and find the effective Landau-Lifshitz Lagrangians to four loops. We extend to four-loops λ-tilde4 the computations of 1/J and 1/J2 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 λ-tilde4. We compare also the 1/J and 1/J2 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