A rederivation of the conformal anomaly for spin-${\frac{1}{2}}$

Abstract

We rederive the conformal anomaly for spin-1 2 fermions by a genuine Feynman graph calculation, which has not been available so far. Although our calculation merely confirms a result that has been known for a long time, the derivation is new, and thus furnishes a method to investigate more complicated cases (in particular concerning the significance of the quantum trace of the stress tensor in non-conformal theories) where there remain several outstanding and unresolved issues.

Axial gravity: a non-perturbative approach to split anomalies

“…Overview of the literature There exists a vast literature on even trace anomalies in 4d, mostly old but also recent [33][34][35], denoting a renewed interest in the subject. The literature on the odd parity trace anomaly in 4d (still in a settling phase) consists of [1][2][3][36][37][38][39][40].…”

Conformal anomaly for non-conformal scalar fields

“…For the perturbative determination of the anomaly (1) we follow the same procedure as in [18] and expand the metric in the usual way…”

“…where · · · 0 refers to the free scalar expectation value (below we will often drop the subscript 0 when it is obvious what is meant). The evaluation of the Feynman diagrams resulting from this expansion is completely analogous to the calculation performed in [18] to which we refer for further technical details.…”

“…This is in marked contrast to the coefficients of the E 4 and C 2 anomalies at O(h 2 ) which exhibit no such prescription dependence. This is the reason why the nice trick that allows the c-coefficient to be determined from the O(h) computation [2] (see also [18]) no longer works for non-conformal theories. Consequently for ξ = 1 6 the determination of the a and c coefficients requires a calculation at O(h 2 ).…”

“…In this paper we focus on the investigation of the conformal anomaly and its significance for the specific example of the non-conformal scalar field. For this we follow a recent recalculation of the anomaly for spin-1 2 fields [18], which is based on an evaluation of Feynman diagrams largely analogous to (though much more involved than) the textbook derivation of the axial anomaly in gauge theories. The spin-0 field coupled to a background gravitational field represents the simplest example in which to consider non-conformal deformations, and has the added advantage that there is a tunable parameter ξ, such that for one special value of ξ the theory becomes conformal.…”