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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_{}^{J_X L_X P_X}= }	defining an amplitude...
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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: $\pi ^{0}\oplus (\pi ^{+}\oplus \pi ^{-})\rightarrow \omega$