The finite effective action in the non-linear sigma model with torsion to two-loop order

Abstract

Quantum corrections to the two-dimensional bosonic non-linear sigma model with torsion are investigated using a recently developed regularization technique called Operator Regularization. This technique does not entail modification of the original action in order to compute such corrections and so does not manifestly violate the conformal symmetry of the classical non-linear sigma model (in contrast to other methods as dimensional regularization and Pauli-Villars). We evaluate the scale-dependent part of the effective action to order ħ 2 and compute the associated β-function. We repeat this analysis using Dimensional Regularization. We find for non-zero torsion that the Renormalization Group Equation is consistent but the β-function derived from it in both Operator Regularization and Dimensional Regularization differs from that derived from the poles using Dimensional Regularization. The vanishing of this β-function is not consistent with the equations of motion that follow from a string effective action. In the context of a “double background” formalism we show that a finite renormalization may be performed in both procedures, reconciling the discrepancy between them but making the β-function dependent on an arbitrary parameter. Although this parameter may be chosen to restore consistency with the equations of motion that follow from a string effective action, it is insufficient to resolve the aforementioned discrepancy between the Renormalization Group Equations and the pole equations in Dimensional Regularization. This problem may be ameliorated by introducing a potential for the regularization of infrared divergences; however in this case the Renormalization Group Equation is inconsistent. If the torsion is zero then we find consistency in all sectors up to the inclusion of terms proportional to the β-function of order ħ.