Abstract
This article deals with two main topics. One is odd parity trace anomalies in Weyl fermion theories in a 4d curved background, the second is the introduction of axial gravity. The motivation for reconsidering the former is to clarify the theoretical background underlying the approach and complete the calculation of the anomaly. The reference is in particular to the difference between Weyl and massless Majorana fermions and to the possible contributions from tadpole and seagull terms in the Feynman diagram approach. A first, basic, result of this paper is that a more thorough treatment, taking account of such additional terms and using dimensional regularization, confirms the earlier result. The introduction of an axial symmetric tensor besides the usual gravitational metric is instrumental to a different derivation of the same result using Dirac fermions, which are coupled not only to the usual metric but also to the additional axial tensor. The action of Majorana and Weyl fermions can be obtained in two different limits of such a general configuration. The results obtained in this way confirm the previously obtained ones.
Highlights
The second topic is motivated as follows
One is odd parity trace anomalies in Weyl fermion theories in a 4d curved background, the second is the introduction of axial gravity
One is odd parity trace anomalies in chiral fermion theories in a 4d curved a e-mail: bonora@sissa.it b e-mail: mcvitan@phy.hr c e-mail: pprester@phy.uniri.hr d e-mail: duarte763@gmail.com e e-mail: sgiaccari@phy.hr f e-mail: tstember@phy.hr background, the second is the introduction of axial gravity
Summary
The second topic is motivated as follows. It is well known that in anomaly calculations the functional integral measure plays a basic role. Let us consider the case in which there is no quantum number appended to the fermions The reason why they are sometimes considered as a unique object is due, we think, to the fact that we can establish a one-to-one correspondence between the components of a Weyl spinor and those of a Majorana spinor in such a way that the Lagrangian ( in two-component notation) looks the same. We intend to return more punctually on this issue, let us point out for the time being that using a Dirac fermion path integration measure amounts to introducing in the game both chiralities, even though formally the action is declared to be the Weyl one. Most important, as already pointed out, in the quantum theory a crucial role is played by the functional measure, which is very likely to be different for Weyl and Majorana fermions. The latter is 0 for a massless Majorana fermion, while it equals the Pontryagin density for a Weyl fermion (the even parity trace anomaly is the same for both)
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