Dirac, Majorana, Weyl in 4d

TL;DR

This review explores the fundamental properties of Dirac, Weyl, and Majorana spinors in four dimensions, emphasizing differences, effective actions, and anomalies, including a notable relation between chiral and trace anomalies.

Contribution

It provides a detailed comparison of massless and massive spinors, clarifies their effective actions with gauge fields, and revisits anomalies with insights on regularization independence and their interrelations.

Findings

Regularization independence in perturbative anomalies

Distinct behaviors of massless and massive spinors

Connection between chiral and trace anomalies

Abstract

This is a review of some elementary properties of Dirac, Weyl and Majorana spinors in 4d. We focus in particular on the differences between massless Dirac and Majorana fermions, on one side, and Weyl fermions, on the other. We review in detail the definition of their effective actions, when coupled to (vector and axial) gauge fields, and revisit the corresponding anomalies using the Feynman diagram method with different regularizations. Among various well known results we stress in particular the regularization independence in perturbative approaches, while not all the regularizations fit the non-perturbative ones. As for anomalies, we highlight in particular one perhaps not so well known feature: the rigid relation between chiral and trace anomalies.