Reexamining the photon polarization in $B\to K\pi\pi\gamma$

We reexamine, update and extend a suggestion we made fifteen years ago for measuring the photon polarization in $b \to s\gamma$ by observing in $B \to K\pi\pi\gamma$ an asymmetry of the photon with respect to the $K\pi\pi$ plane. Asymmetries are calculated for different charged final states due to intermediate $K_1(1400)$ and $K_1(1270)$ resonant states. Three distinct interference mechanisms are identified contributing to asymmetries at different levels for these two kaon resonances. For $K_1(1400)$ decays including a final state $\pi^0$ an asymmetry around $+30\%$ is calculated, dominated by interference of two intermediate $K^*\pi$ states, while an asymmetry around $+10\%$ in decays including final $\pi^+\pi^-$ is dominated by interference of $S$ and $D$ wave $K^*\pi$ amplitudes. In decays via $K_1(1270)$ to final states including a $\pi^0$ a negative asymmetry is favored up to $-10\%$ if one assumes $S$ wave dominance in decays to $K^*\pi$ and $K\rho$, while in decays involving $\pi^+\pi^-$ the asymmetry can vary anywhere in the range $-13\%$ to $+24\%$ depending on unknown phases. For more precise asymmetry predictions in the latter decays we propose studying phases in $K_1 \to K^*\pi, K\rho$ by performing dedicated amplitude analyses of $B\to J/\psi(\psi') K\pi\pi$. In order to increase statistics in studies of $B\to K\pi\pi\gamma$ we suggest using isospin symmetry to combine in the same analysis samples of charged and neutral $B$ decays.