Interacting rigid strings with long range order: beyond the Gaussian approximation

Recently we have shown that a phase transition occurs in the leading and sub-leading approximation of the large N limit in rigid strings coupled to long range Kalb-Ramond interactions. The disordered phase is essentially the Nambu-Goto-Polyakov string theory while the ordered phase is a new theory. The non-local potential for the string generated by the Kalb-Ramond (KR) interactions is not quadratic in the string and an additional approximation in all of the large N analysis mentioned was that only the quadratic piece of the potential was retained. In this letter we study the first non-trivial cubic effect of the KR potential and we set up the rules for perturbation theory in large N. We show that the theory is super-renormalizable and obtain the renormalized two point function. Consequently, we show that the quadratic approximation is self-consistent in that the region of the renormalized couplings of the extrinsic curvature and the KR couplings contains the phase with long range order. The result suggests that the phase transition should survive beyond the Gaussian approximation in large N.