Amplitudes for the Exotic b1π Decay

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