Abstract
We revisit the computation of the trace anomaly for Weyl fermions using dimensional regularization. For a consistent treatment of the chiral gamma matrix γ* in dimensional regularization, we work in n dimensions from the very beginning and use the Breitenlohner-Maison scheme to define γ*. We show that the parity-odd contribution to the trace anomaly vanishes (for which the use of dimension-dependent identities is crucial), and that the parity-even contribution is half the one of a Dirac fermion. To arrive at this result, we compute the full renormalized expectation value of the fermion stress tensor to second order in perturbations around Minkowski spacetime, and also show that it is conserved.
Highlights
The axial vector potential in gauge theory
We revisit the computation of the trace anomaly for Weyl fermions using dimensional regularization
We show that the parity-odd contribution to the trace anomaly vanishes, and that the parity-even contribution is half the one of a Dirac fermion
Summary
Let us introduce left-handed Weyl fermions, which are four-component fermions satisfying ψ. Where γ∗ is the chiral γ matrix satisfying γ∗2 = 1, such that the projectors P± are idempotent: P±2 = P±. In a curved n-dimensional space, the action for Weyl fermions reads. Where in the second expression we have explicitly displayed the projectors. In this expression, the covariant derivative for the fermion includes the spin connection ω,. The curved-space γ matrices are obtained as usual from the constant flat-space ones by introducing the vielbein, eμa, namely as γμ ≡ gμνeνaηabγb. From the action (2.2), one can compute the associated symmetric stress tensor [34] (with the projectors explicitly displayed). Rμν ρσ γρσ ψ [32]), classically this tensor is conserved and its trace vanishes on-shell, since the action for a Weyl fermion is conformally invariant. As we will see the situation can change at the quantum level, giving rise to the trace anomaly
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