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
$\left[P_{X}(-)^{J_{X}+1+\epsilon }e^{2i\alpha }\left({\begin{array}{cc|c}J_{b1}&L_{X}&J_{X}\\m_{b1}&m_{X}&-1\end{array}}\right)+\left({\begin{array}{cc|c}J_{b1}&L_{X}&J_{X}\\m_{b1}&m_{X}&+1\end{array}}\right)\right]$	resonance helicity sum: ε=0 (1) for x (y) polarization; </math>P_X</math> is the parity of the resonance
$\left({\frac {1+(-)^{\epsilon }\eta }{4}}\right)$	polarization term: η is the polarization fraction
$k^{L_{X}}q^{J_{b1}}$	k, q are breakup momenta for the resonance and isobar, respectively
$\left({\begin{array}{cc|c}I_{b1}&I_{\pi }&I_{X}\\I_{z\pi ^{+}}&I_{z\pi ^{-}}&I_{z\pi ^{+}}+I_{z\pi ^{-}}\end{array}}\right)$	Clebsch-Gordan coefficients for isospin sum $b_{1}\oplus \pi ^{-}\rightarrow X$
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: $\pi ^{0}\oplus (\pi ^{+}\oplus \pi ^{-})\rightarrow \omega$