Photon polarization in radiativeBdecays

We study decay distributions in $\overrightarrow{B}K\pi \pi \gamma ,$ combining contributions from several overlapping resonances in a $K\pi \pi$ mass range near 1400 MeV, $1^{+}$ $K_1\mathrm{}\left(1400\right),$ $2^{+}$ $K_2^{*}\mathrm{}\left(1430\right)\mathrm{}$ and $1^{-}$ $K^{*}\mathrm{}\left(1410\right).$ A method is proposed for using these distributions to determine a photon polarization parameter in the effective radiative weak Hamiltonian. This parameter is measured through an up-down asymmetry of the photon direction relative to the $K\pi \pi$ decay plane in the $K_{\mathrm{res}}$ frame. We calculate a dominant up-down asymmetry of $0.33\pm 0.05$ from the $K_1\mathrm{}\left(1400\right)\mathrm{}$ resonance, which can be measured with about ${10}^8$ $B\overline{B}$ pairs, thus providing a new test for the standard model and a probe for some of its extensions.