Abstract
We provide the renormalization constants of the axial current operators, both singlet and non-singlet, in dimensional regularization up to four-loop order in (QCD), determined using the off-shell Ward-Takahashi identity for an axial current with a non-anticommuting γ5. A possible application of the result for the singlet axial current operator is the extraction of the non-decoupling mass logarithms in the axial quark form factors.
Highlights
We provide the renormalization constants of the axial current operators, both singlet and non-singlet, in dimensional regularization up to four-loop order in (QCD), determined using the off-shell Ward-Takahashi identity for an axial current with a nonanticommuting γ5
The missing O(αs3) finite part was completed in ref. [39] where a derivative-free projector was composed in order to efficiently project out the non-vanishing “anomalous” form factor of the famous vector-vector-axial-vector (VVA)-amplitude [3, 40], i.e. the matrix element of the axial current operator between the vacuum and a pair of external gluons, at zero-momentum
We present our final perturbative results for the renormalization constants defined in eqs. (2.2), (2.3) for axial currents, as well as the ZF J in eq (2.9), at the four-loop order in with nf massless quarks
Summary
We specify the notations and conventions for the quantities considered in the rest of this article. The non-singlet axial current J5a,,nμs is conserved, up to the classically-expected mass term 2mψiγ5taψ which vanishes for massless quark fields, and is non-anomalous. This property is sufficient to ensure that J5a,,nμs should remain un-renormalized if one had used an anticommuting γ5, to the non-renormalization of the vector current [Jμ]R = ψBγμψB. The γ5’s anticommutativity is effectively restored for the non-singlet axial current [15, 16, 41] This implies that the renormalization constant Zns ≡ Znfs Znmss, as a whole, has a vanishing anomalous dimension, namely μ2 d ln Zns dμ. The matrix of anomalous dimensions of these two renormalized operators is defined by d d ln μ2
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