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

Arroyo-Ureña, Hernández-Tomé, López-Castro, Roig, and Rosell (AHLRR) take the structure-dependent (SD) amplitude for $\tau \to P{\nu }_{\tau }\gamma$ from the resonance chiral theory result of Guo and Roig (GR10), obtaining $\delta r_{SD}=+0.15\mathrm{\%}$ 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: $\delta r_{SD}$ shifts from $+0.150\mathrm{\%}$ to $+0.146\mathrm{\%}$ in the pion channel and from $+0.18\mathrm{\%}$ to about $+0.16\mathrm{\%}$ in the kaon channel.