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

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}.