Clarifying some remaining questions in the anomaly puzzle

Abstract

We discuss several points that may help to clarify some questions that remain about the anomaly puzzle in supersymmetric theories. In particular, we consider a general N = 1 supersymmetric Yang-Mills theory. The anomaly puzzle concerns the question of whether there is a consistent way to put the R-current and the stress tensor in a single supercurrent, even though in the classical theory they are in the same supermultiplet. As is well known, the classically conserved supercurrent bifurcates into two supercurrents having different anomalies in the quantum regime. The most interesting result we obtain is an explicit expression for the lowest component of one of the two supercurrents in 4-dimensional spacetime, namely the supercurrent that has the energy-momentum tensor as one of its components. This expression for the lowest component is an energy-dependent linear combination of two chiral currents, which itself does not correspond to a classically conserved chiral current. The lowest component of the other supercurrent, namely, the R-current, satisfies the Adler-Bardeen theorem. The lowest component of the first supercurrent has an anomaly that we show is consistent with the anomaly of the trace of the energy-momentum tensor. Therefore, we conclude that there is no consistent way to put the R-current and the stress tensor in a single supercurrent in the quantized theory. We also discuss and try to clarify some technical points in the derivations of the two-supercurrents in the literature. These latter points concern the significance of infrared contributions to the NSVZ β-function and the role of the equations of motion in deriving the two supercurrents.

On the trace anomaly and the anomaly puzzle in $ \mathcal{N} = 1 $ pure Yang-Mills

“…We do not perform a detailed study on this point, but the existence of the two different supercurrents cannot be confirmed unless the possibility (48) is excluded. (However, the idea of two supercurrents becomes important when matter fields are included in the theory [11,12,33]. )…”

“…(28), which is completely determined in UV region. 12 Then the question of which beta function should appear in the trace anomaly comes back to us. This is precisely the problem we have studied in this paper.…”

“…(See also Ref. [12] for a related work.) More explicitly, let us consider a theory with matter chiral fields Φ r which transform in the representation r of the gauge group.…”

Regularization independent analysis of the origin of two loop contributions to N=1 Super Yang–Mills beta function

“…As a result, the beta function corrections are not constrained by the Adler Bardeen theorem. However, some controversies remain [8].…”

Perturbative c-theorem in d-dimensions

“…In higher orders of perturbation theory, there is a notorious problem called the anomaly puzzle[44]. See Refs [45,46,47,48,49,50]. and references therein for discussions on this problem.…”