Towards a Systematic method of implicit renormalization: chiral theories and higher orders

Abstract

In the first part of the thesis, we study the performance of implicit methods in chiral theories. The fact that implicit methods work at fixed dimension make them good candidates for the convenient renormalization of chiral gauge theories. However, we show that, under very mild assumptions (which hold in these methods) there is an unavoidable conflict between the preservation of gauge invariance and the validity of dimension-specific identities that are related to the standard properties of the 5 matrix. As a consequence, the implicit methods of interest present exactly the same problems with chiral theories as the dimensional methods. The original formulations of these methods lead in fact to inconsistencies in dealing with the " antisymmetric tensor and the 5 matrix. To remediate this, we supplement the implicit methods with additional rules that render the results unambiguous. In this way, they can be safely used in chiral gauge theories. The price to pay is giving up standard Fierz identities and some of the familiar properties of the 5. We perform explicit one-loop calculations in chiral theories with FDR to assess the performance of the improved methods. The second part of the thesis deals with multi-loop calculations. We concentrate on FDR. As mentioned above, in the absence of explicit counterterms the unitarity and locality of the renormalized theory is not guaranteed a priori. In fact, the naive formulation of FDR was soon found to be inconsistent with the counterterm structure of renormalizable theories. This was solved in explicit two-loop calculations by incorporating into the method a “subprescription” that enforces subintegration consistency. This works in known examples, but still, it is a correction a posteriori and it is not clear if it will work as such for all Feynman diagrams, especially at higher loops or in the presence of infrared singularities. Our proposal is to enforce the essential properties by organizing the renormalization of diagrams and subdiagrams according to Zimmermann’s Forest Formula. To do this, we define a subtraction operator in the context of FDR. This determines the method. We analyze different definitions and test them in explicit two-loop calculations. They are all shown to respect the Ward identities. However, some of them present problems in reproducing the known beta functions, which reflect wrong non-local parts in the renormalized amplitudes. Finally, we select the simplest definition and check that it leads to all the desirable properties. In fact, we show that the Forest Formula with the selected operator automatically generates the same “extra-extra” terms as the FDR subprescription in all the examples we study. The developments of the two chapters lead, as the main result of the thesis, to the precise definition of a systematic implicit method that respects gauge invariance and other basic properties of chiral and non-chiral quantum field theories, at least at the two-loop level. Based on our analysis, we believe that the very same method can also be used successfully, without any further refinement, at arbitrary order. But a rigorous proof to all orders goes beyond the scope of this work.