Comment on the sign of the pseudoscalar pole contribution to the muon g-2

TL;DR

This paper corrects a sign error in the pseudoscalar pole contribution to muon g-2 calculations, aligning previous results with recent findings and clarifying the impact of algebraic manipulation issues.

Contribution

It identifies and rectifies a sign error caused by FORM's epsilon-tensor definition, improving the accuracy of muon g-2 theoretical estimates.

Findings

Corrected the sign of the pseudoscalar pole contribution.

Confirmed agreement with recent independent results.

Highlighted the importance of tensor algebra conventions in calculations.

Abstract

We correct the error in the sign of the pseudoscalar pole contribution to the muon g-2, which dominates the O(alpha^3) hadronic light-by-light scattering effect. The error originates from our oversight of a feature of the algebraic manipulation program FORM which defines the epsilon-tensor in such a way that it satisfies the relation epsilon_{mu_1 mu_2 mu_3 mu_4} epsilon_{nu_1 nu_2 nu_3 nu_4} eta^{mu_1 nu_1} eta^{mu_2 nu_2} eta^{mu_3 nu_3} eta^{mu_4 nu_4} = 24, irrespective of space-time metric. To circumvent this problem, we replaced the product epsilon_{mu_1 mu_2 mu_3 mu_4} epsilon_{nu_1 nu_2 nu_3 nu_4} by - eta_{mu_1 nu_1} eta_{mu_2 nu_2} eta_{mu_3 nu_3} eta_{mu_4 nu_4} \pm cdots in the FORM-formatted program, and obtained a positive value for the pseudoscalar pole contribution, in agreement with the recent result obtained by Knecht {\it et al}.