Comment on "Radiative corrections to tau -> pi(K) nu_tau[gamma]: A reliable new physics test"

Abstract

Arroyo-Ureña, Hernández-Tomé, López-Castro, Roig, and Rosell (AHLRR) take the structure-dependent (SD) amplitude for $τ\to P ν_τγ$ from the resonance chiral theory result of Guo and Roig (GR10), obtaining $δr_{SD} = +0.15\%$ in the pion channel. The convention dictionary in footnote 2 of GR10, repeated in footnote 1 of AHLRR, has the wrong sign in its axial entry: the printed amplitudes imply $F_A^{DF} = -2\sqrt{2}\, m_P F_A^{GR}$, not $+2\sqrt{2}\, m_P F_A^{GR}$. With the corrected dictionary, the GR10 amplitude agrees with the $O(p^4)$ chiral prediction in the axial sector but carries the opposite sign to the chiral anomaly in the vector sector; the relative sign fixed in the Decker-Finkemeier addendum is confirmed in both sectors. The identification in footnote 6 of AHLRR of their value with the pre-addendum result rests on the erroneous dictionary and does not hold. The numerical impact is small: $δr_{SD}$ shifts from $+0.150\%$ to $+0.146\%$ in the pion channel and from $+0.18\%$ to about $+0.16\%$ in the kaon channel.