Amplitudes for the Exotic b1π Decay

Let's begin with the amplitude for decay of a state X with some $J_{X},M_{X}$ quantum numbers:

$\langle \Omega _{X}0\lambda _{b_{1}}|U_{X}|J_{X}m_{X}\rangle =\langle \Omega _{X}0\lambda _{b_{1}}|J_{X}m_{X}L_{X}s_{b_{1}}\rangle \langle J_{X}m_{X}L_{X}s_{b_{1}}|U_{X}|J_{X}m_{X}\rangle$

OLD

$A_{}^{J_{X}L_{X}P_{X}}=$	defining an amplitude...
$\sum \limits _{m_{X}=-L_{X}}^{L_{X}}\sum \limits _{m_{b1}=-J_{b_{1}}}^{J_{b_{1}}}\sum \limits _{m_{\omega }=-J_{\omega }}^{J_{\omega }}D_{m_{X}m_{b_{1}}}^{L_{X}*}(\theta _{X},\phi _{X},0)D_{m_{b_{1}}m_{\omega }}^{J_{b_{1}}*}(\theta _{b_{1}},\phi _{b_{1}},0)$	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
$\left({\begin{array}{cc|c}I_{b_{1}}&I_{\pi }&I_{X}\\I_{zb_{1}^{+}}&I_{z\pi ^{-}}&I_{zb_{1}^{+}}+I_{z\pi ^{-}}\end{array}}\right)$	Clebsch-Gordan coefficients for isospin sum $b_{1}\oplus \pi ^{-}\rightarrow X$
$\sum \limits _{L_{b_{1}}=0}^{2}\sum \limits _{m_{L_{b_{1}}}=-L_{b_{1}}}^{L_{b_{1}}}\sum \limits _{L_{\pi ^{+}\pi ^{-}},L_{\omega }=1,3}\sum \limits _{m_{\pi ^{+}\pi ^{-}}=-L_{\pi ^{+}\pi ^{-}}}^{L_{\pi ^{+}\pi ^{-}}}u^{L_{\omega }}v^{L_{\pi ^{+}\pi ^{-}}}$
$D_{m_{\omega }m_{\pi ^{+}\pi ^{-}}}^{J_{\omega }*}(\theta _{\omega },\phi _{\omega },0)Y_{m_{\pi ^{+}\pi ^{-}}}^{L_{\pi ^{+}\pi ^{-}}}(\theta _{\rho },\phi _{\rho })$	two-stage $\omega (J_{\omega }^{PC}=1^{--})$ breakup angular distributions, currently modeled as $L_{\omega \rightarrow \pi ^{0}+\rho }=0;L_{\rho \rightarrow \pi ^{+}+\pi ^{-}}=1=L_{\pi ^{+}\pi ^{-}}$
$\left({\begin{array}{cc|c}J_{\omega }&L_{b_{1}}&J_{b_{1}}\\m_{\omega }&m_{L_{b_{1}}}&m_{b_{1}}\end{array}}\right)\left({\begin{array}{cc|c}L_{\omega }&L_{\pi ^{+}\pi ^{-}}&J_{\omega }\\0&m_{\pi ^{+}\pi ^{-}}&m_{\omega }\end{array}}\right)$	angular momentum sum Clebsch-Gordan coefficients for b1 and ω decays.
$\left({\begin{array}{cc|c}I_{\pi }&1&0\\I_{\pi ^{0}}&0&0\end{array}}\right)\left({\begin{array}{cc|c}I_{\pi }&I_{\pi }&1\\I_{z\pi ^{+}}&I_{z\pi ^{-}}&0\end{array}}\right)$	Clebsch-Gordan coefficients for isospin sums: $\pi ^{0}\oplus (\pi ^{+}\oplus \pi ^{-})\rightarrow \omega$