Loriano Bonora

This is a review paper intended to illustrate a few critical issues concerning Dirac, Weyl, and Majorana fermions and their differences. The first part consists of a basic introduction to fermions, with further detail on fermions in 4d, focusing in particular on what differentiates the three types of fermions: chirality, helicity, mass, field equations, properties under discrete symmetry transformations, actions, and observables. On this basis, we tackle a series of challenging and sometimes controversial problems: the non-existence of Weyl fermion propagators, and ways to circumvent it; the regularizations for Weyl fermion amplitudes, in particular the appropriateness of using PV regularization; the definition of a functional integral for a Weyl fermion; the difference between Weyl and massless Majorana fermions; and the difference between Dirac and Majorana mass terms. Most notably, we come to the conclusion that the method of replacing a non-existing Weyl fermion propagator with a massless Dirac one, although it may yield, in some cases, correct results, is flawed at the very origin by a logical loophole. We then show what is the correct way to proceed in this case. We also consider the topic of applying a Wick rotation at the classical action level for Weyl fermions and conclude that this procedure leads to a nonequivalent theory. Finally, although anomalies are not the central focus here, we have deemed it useful to summarily review the relation, and its cohomological basis, that exists between the lack of a Weyl fermion propagator (non-invertibility of the Weyl–Dirac operator) and the appearance of dangerous anomalies in the theory.

This paper is a bottom-up attempt to incorporate the standard model and general relativity in a unique quantum field theory. In particular, the tentative model presented here is free of chiral gauge and gravitational anomalies that appear in the divergence of currents and in the divergence and trace of the energy–momentum tensor when SM matter couples to gravity. The fermion spectrum is composed of two multiplets: the SM (left) multiplet and a mirror copy (right) with opposite handedness. The right multiplet is interpreted as describing the dark matter world. The natural symmetry of the theory is enlarged to incorporate Weyl invariance by introducing one or more dilaton fields. After the cosmological and theoretical motivations, the necessary formalism is introduced for algebraic renormalization: gauge fixings, ghosts, propagators, vertices and their interplay in guaranteeing the conditions for convergence of the subtracted amplitudes according to the BPHZL scheme, the Slavnov–Taylor identity, and the relevant enlarged BRST symmetry. The corresponding (conformal) cohomology is analyzed and found to be trivial: there are no non-trivial even trace anomalies in theories with dilatons, but there are plenty of trivial ones that require corresponding counterterms in the effective action. It is shown that such counterterms can play an important role in freeing the theory from unphysical particles.

This is a review paper intended to illustrate a few critical issues concerning Dirac, Weyl and Majorana fermions and their differences. The first part consists in basic introductions to fermions, and more in detail to fermions in 4d, focusing in particular in what differentiate the three type of fermions: chirality, helicity, mass, field equations, properties under discrete symmetry transformations, Actions, Observables. On this basis we tackle a series of challenging and sometime controversial problems: the non-existence of Weyl fermion propagators, and the ways to circumvent it; the regularizations for Weyl fermion amplitudes, in particular the appropriateness of using the PV regularization; the definition of a functional integral for Weyl fermion, the difference between Weyl and massless Majorana fermions and the difference between the Dirac and Majorana mass terms. Most notably we come to the conclusion that the trick of replacing the non-existing Weyl fermion propagator with a massless Dirac one, although it may yield in some cases correct results, is flawed at the very origin by a logical loophole. We then show which is the correct way to proceed in this case. We consider also the topic of applying the Wick rotation at the classical Action level for Weyl fermions and conclude that this procedure leads to a nonequivalent theory. Finally, although anomalies are not the central focus here, we have deemed it useful to summarily review the relation, and its cohomological basis, that exists between the lack of a Weyl fermion propagator (non-invertibility of the Weyl-Dirac operator) and the appearance of dangerous anomalies in the theory.

This paper is a bottom up attempt to incorporate the standard model and general relativity in a unique quantum field theory. The tentative model presented here in particular is free of chiral gauge and gravitational anomalies that appear in the divergence of currents, and in the divergence and trace of the energy-momentum tensor when the SM matter couples to gravity. The fermion spectrum is composed of two multiplets, the SM (left) multiplet and a mirror copy (right) with opposite handedness. The right multiplet is interpreted as describing the dark matter world. The natural symmetry of the theory is enlarged to incorporate also Weyl invariance, by introducing one or more dilaton fields. After the cosmological and theoretical motivations, the necessary formalism is introduced for algebraic renormalization: gauge fixings, ghosts, propagators and vertices and their interplay in garanteeing the conditions for convergence of the subtracted amplitudes according to the BPHZL scheme, the Slavnov-Taylor identity and the relevant enlarged BRST symmetry. The corresponding (conformal) cohomology is analyzed and found to be trivial: there are no non-trivial even trace anomalies in theories with dilatons, but there are plenty of trivial ones, which require corresponding counterterms in the effective action. It is shown that such counterterms can play an important role in freeing the theory of unphysical particles.

This is a review/research paper on anomalies applied in a bottom–up approach to standard model and gravity. It is divided into two parts. The first consists of a proper review of anomalies in quantum field theories. Anomalies are analyzed according to three different methods: a perturbative one based on a Feynman diagram, a non-perturbative one relying on the Schwinger–DeWitt approach, and, third, one hinging on the Atiyah–Singer family’s index theorem. The three methods are applied both to chiral gauge anomalies and trace anomalies. The fundamental distinction, which our presentation leads to, is between obstructive (O) and non-obstructive (NO) anomalies. The former is tied to the non-existence of fermion propagators, which fatally maim the corresponding theory. In the second part, we apply this analysis to the SM and a variety of its extensions, which are immersed in a gravitational background, and we find that they are all plagued by a residual chiral trace anomaly. To completely eliminate all kinds of dangerous anomalies in SM-like theories, we propose a somewhat unconventional scheme and exemplify it by means of an explicit model. The latter is a left–right symmetric model. We embed it in a Weyl geometry to render it a conformal invariant. We then deal with some of its quantum aspects, particularly its even (NO) trace anomalies and the means to preserve its conformal invariance at the quantum level. We briefly review renormalization and unitarity in the framework of similar models discussed in the existing literature. Finally, we present a possible (conjectural) application of the model to describe the junction between cosmology and quantum field theory.

This ia a review of anomalies applied to a bottom-up approach to standard model and gravity. It is divided in two parts. The first consists in a review proper of anomalies in quantum field theories. Anomalies are analyzed according to three different methods: a perturbative one based on Feynman diagram, a non-perturbative one relying on the Schwinger-DeWitt approach and, third, the one hinging on the Atiyah-Singer family's index theorem. The three methods are applied both to chiral gauge anomalies and trace anomalies. The fundamental distinction that our presentation leads to is between obstructive (O) and non-obstructive (NO) anomalies. The former are tied to the non-existence of fermion propagators, which fatally maim the corresponding theory. In the second part we apply this analysis to the SM and various of its extensions immersed in a gravitational background, and find that they all are plagued by a residual chiral trace anomaly. To completely eliminate all kind of dangerous anomalies in SM-like theories we propose a somewhat unconventional scheme, and exemplify it by means of an explicit model. The latter is a left-right symmetric model. We embed it in a Weyl geometry to render it conformal invariant. We then deal with some of its quantum aspects, in particular its even (NO) trace anomalies and the means to preserve its confomal invariance at the quantum level. We briefly review renormalization and unitarity in the framework of similar models discussed in the existing literature. Finally we present a possible (conjectural) application of the model to describe the junction between cosmology and quantum field theory.

Weyl 0- and 1-cocycles of canonical dimension 6 in six dimensions, which were computed earlier in Bonora et al (1986 Class. Quantum Grav. 3 635), are recalculated from scratch. The analysis yields five Weyl invariants (0-cocycles), instead of four, and the same four non-trivial 1-cocycles (possible trace anomalies), like in that reference (up to the correction of one typo).