Amplitudes for the Exotic b1π Decay

A_{}^{J_X L_X P_X}

defining an amplitude...
$\sum \limits _{m_{X}=-L_{X}}^{L_{X}}\sum \limits _{m_{b1}=-J_{b1}}^{J_{b1}}Y_{m_{X}}^{L_{X}}(\theta _{X},\phi _{X})D_{m_{b1}n_{b1}}^{J_{b1}*}(\theta _{b1},\phi _{b1},0)$	angular distributions two-body X and $b_{1}(J_{b_{1}}^{PC}=1^{+-})$ decays
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 \left[ P_X(-)^{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: ε=0 (1) for x (y) polarization; </math>P_X</math> is the parity of the resonance
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 \left(\frac{1+(-)^\epsilon \eta}{4}\right) }	polarization term: η is the polarization fraction
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 k^{L_X} q^{J_{b1}} }	k, q are breakup momenta for the resonance and isobar, respectively
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 \left(\begin{array}{cc|c} I_{b1} & 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 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 \oplus \pi^- \rightarrow X}
Failed to parse (syntax error): {\displaystyle \sum\limits_{L_{b1}=0}^{2} \sum\limits_{m_{L_{b1}}=-L_{b1}}^{L_{b1}} \sum\limits_{m_\omega}=-J_\omega}^{J_\omega} \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 $L_{\omega \rightarrow \pi ^{0}-\rho }=0;L_{\rho \rightarrow \pi ^{+}+\pi ^{-}}=1=s_{\rho }$
$\left({\begin{array}{cc|c}J_{\omega }&L_{b1}&J_{b1}\\m_{\omega }&m_{L_{b1}}&m_{b1}\end{array}}\right)\left({\begin{array}{cc|c}1&s_{\rho }&J_{\omega }\\0&\lambda _{\rho }&m_{\omega }\end{array}}\right)$	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: $\pi ^{0}\oplus (\pi ^{+}\oplus \pi ^{-})\rightarrow \omega$