Quantum Stability of the Phase Transition in Rigid QED

Rigid QED is a renormalizable generalization of Feynman’s space-time action characterized by the addition of the curvature of the world line (rigidity). We have recently shown that a phase transition occurs in the leading approximation of the large N limit. The disordered phase essentially coincides with ordinary QED, while the ordered phase is a new theory. We have further shown that both phases of the quantum theory are free of ghosts and tachyons. In this letter, we study the first sub-leading quantum corrections leading to the renormalized mass gap equation. Our main result is that the phase transition does indeed survive these quantum fluctuations. ∗ E-Mail address:(moustafa@physunc. phy.uc.edu IPhase Transition in Rigid Model Of QED Recenlty we proposed a renormalizable generalization of the Feynman spacetime picture of QED [1], [2]. In this picture the dynamical variables are the spacetime position x, μ = 1, 2, ...D of the point particle and the photon field Aμ. The usual Feynman action consists of the arc-length of the world line, the Maxwell action, and the usual point particle-Maxwell coupling. The renormalizable generalization is characterized by the addition of the scale invariant curvature of the world line. The origin of the term rigid refers to the Boltzmann suppression of curved trajectories by the curvature. In a subsequent article [3] we proved a conjecture in [1] and [2] that there is phase transition in rigid QED. We found a critical line in the plane of the Coulomb coupling verses the curvature coupling below which there is a disordered phase and above which is a new ordered strongly coupled phase. The higher derivative nature of rigid QED should cause a serious pause as any higher derivative theory is typically pathological. Indeed, the arc-length plus the curvature term theory has classical runaway solutions which are tachyonic. Whether a higher derivative regulated quantum theory has such pathological behaviour is more subtle and depends on details of the continuum limit. A free scalar field theory on the lattice with spacing 1 Λ , has higher derivatives and ghosts. However these ghosts have mass of order Λ and decouple in the continuum limit as Λ → ∞. In the less trivial case of rigid QED we have shown that the ghosts have mass of order Λ and similarly decouple from the continuum limit. The necessity of the decoupling mechanism is associated with the absence of fine tuning of the curvature and the Coulomb coupling constants. Having phase transition would be of utmost importance because this would imply that the couplings of the theory are fixed by dimensional transmutation in both the disordered and ordered phases [4]. Our proof in [3] 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. In this letter we will prove that the phase transition in our model of rigid QED survives quantum fluctuations and that the quantum loop corrections to the subleading order lead to mass and wave function renormalizations. As in non-linear sigma model [5], mass renormalization is equivalent to charge renormalization. Thus we obtain the renormalization group equation.