Radiative resonance couplings in $gammapitopipi$

Studies of the reaction $gammapitopipi$, in the context of the ongoing Primakoff program of the COMPASS experiment at CERN, give access to the radiative couplings of the $rho(770)$ and $rho_3(1690)$ resonances. We provide a vector-meson-dominance estimate of the respective radiative width of the $rho_3$, $Gamma_{rho_3topigamma}=48(18)$ keV, as well as its impact on the $F$-wave in $gammapitopipi$. For the $rho(770)$, we establish the formalism necessary to extract its radiative coupling directly from the residue of the resonance pole by analytic continuation of the $gammapitopipi$ amplitude to the second Riemann sheet, without any reference to the vector-meson-dominance hypothesis.