Amplitudes for the Exotic b1π Decay

$\sum \limits _{m_{X}=-L_{X}}^{L_{X}}\sum \limits _{m_{b1}=-J_{b1}}^{J_{b1}}Y_{m_{X}}^{L_{X}}(\theta _{X},\phi _{X})D_{m_{b1}n_{b1}}^{J_{b1}*}(\theta _{b1},\phi _{b1},0)$	angular distributions two-body X and b1 decays
$\left[(-)^{J_{X}+1+\epsilon }e^{2i\alpha }\left({\begin{array}{cc|c}J_{b1}&L_{X}&J_{X}\\m_{b1}&m_{X}&-1\end{array}}\right)+\left({\begin{array}{cc|c}J_{b1}&L_{X}&J_{X}\\m_{b1}&m_{X}&+1\end{array}}\right)\right]$	resonance helicity sum
$\left({\frac {1+(-)^{\epsilon }\eta }{4}}\right)$	polarization term: ε=0 (1) for x (y) polarization; η is the polarization fraction
$k^{L_{X}}q^{J_{b1}}$	k, q are breakup momenta for the resonance and isobar, respectively
$\left({\begin{array}{cc|c}I_{b1}&1&I_{X}\\I_{z\pi ^{+}}&I_{z\pi ^{-}}&I_{z\pi ^{+}}+I_{z\pi ^{-}}\end{array}}\right)$	Clebsch-Gordan coefficients for isospin sum $b_{1}\oplus \pi ^{-}\rightarrow X$
$\sum \limits _{L_{b1}=0}^{2}\sum \limits _{m_{L_{b1}}=-L_{b1}}^{L_{b1}}D_{m_{\omega }\lambda _{\rho }}^{J_{\omega }*}(\theta _{\omega },\phi _{\omega },0)Y_{m_{\rho }}^{s_{\rho }}(\theta _{\rho },\phi _{\rho })$	two-stage ω breakup angular distributions
$\left({\begin{array}{cc|c}s_{\omega }&L_{b1}&J_{b1}\\0&m_{L_{b1}}&m_{b1}\end{array}}\right)\left({\begin{array}{cc|c}1&s_{\rho }&J_{\omega }\\0&\lambda _{\rho }&m_{\omega }\end{array}}\right)$	angular momentum sum Clebsch-Gordan coefficients for b1 and ω decays
$\sum \limits _{I_{\rho }=0}^{1}\sum \limits _{I_{z\rho }=-I_{\rho }}^{I_{\rho }}\left({\begin{array}{cc|c}1&I_{\rho }&0\\0&I_{z\rho }&0\end{array}}\right)\left({\begin{array}{cc|c}I_{\pi }&I_{\pi }&I_{\rho }\\+1&-1&I_{z\rho }\end{array}}\right)$	Clebsch-Gordan coefficients for isospin sums: $\pi ^{0}\oplus (\pi ^{+}\oplus \pi ^{-})\rightarrow \omega$