Renormalizability and Quantum Stability of the Phase Transition in Rigid String Coupled to Kalb-Ramond Fields II

Recently we have shown that a phase transition occurs in the 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. In this part II letter we study the first sub-leading quantum corrections we started in I. We derive the renormalized mass gap equation and obtain the renormalized critical line of the interacting theory. Our main and final result is that the phase transition does indeed survive quantum fluctuations. ∗ E-Mail address: [email protected] Polyakov [1] has argued that the string theory appropriate to QCD should be one with long range correlations of the unit normal. The Nambu-Goto (NG) string theory is not the correct candidate for large N QCD as it disagrees with it at short distances. The NG string does not give rise to the parton-like behaviour observed in deep inelastic scattering at very high energies. The observed scattering amplitudes have a power fall-off behaviour contrary to the exponential fall-off behaviour of the NG string scattering amplitudes at short distances. The absence of scale and the power law behaviour at short distances suggest that the QCD string must have long range order at very high energies. Pursuing this end, Polyakov considered modifying the Nambu action by the renormalizable scale invariant curvature squared term (rigid strings). The theory closely resembles the two dimensional sigma model where the unit normals correspond to the sigma fields. In the large N approximation, there is no phase transition. Polyakov suggested adding a topological term to produce a phase transition to a region of long range order. In [2], we coupled the rigid strings instead to long range Kalb-Ramond fields. Since spin systems in two dimensions may exhibit a phase transition with the inclusion of long range interactions, it is natural to conjecture likewise for rigid strings with long range Kalb-Ramond fields. Indeed we proved that there is a phase transition to a region of long range order in the large N approximation. Such a theory may therefore be relevant to QCD. Our proof in [2] of the phase transition was based on the leading order approximation in large N, where N is the space-time dimensions. Even though, the large N limit is a successful approximation for non-linear sigma models, and some spin systems it can sometimes lead to an incorrect conclusion. The leading order of the large N approximation is mean field theory which can give incorrect predictions in lower dimensions. For example mean field theory incorrectly gives a phase transition in the one dimensional Ising model. This discrepancy is resolved by carefully examining the sub-leading quantum corrections (loops) where one shows that such quantum corrections in fact destroy the phase transition. Therefore it is crucial to examine the quantum loop corrections to the mass gap, and the critical line of our model of rigid string coupled to long range Kalb-Ramond fields. In this letter we will generalize the analysis we started in I and prove that the phase transition in our model survives quantum fluctuations and that the quantum loop corrections to the sub-leading order lead to mass and wave function renormalizations. Similar prove was given for the phase transition in rigid QED [3] The gauge fixed action of the rigid string [2] coupled to the rank two antisymmetric Kalb-Ramond tensor field φ is [4]: Sgauge−fixed = μ0 ∫ dξρ+ 1 2t0 ∫ dξ[ρ(∂x)+λ(∂ax∂bx−ρδab)]+SK−R (1a)