Amplitudes for the Exotic b1π Decay

Let's begin with the amplitude for decay of a state X with some 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 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...
$\sum \limits _{m_{X}=-L_{X}}^{L_{X}}\sum \limits _{m_{b1}=-J_{b_{1}}}^{J_{b_{1}}}\sum \limits _{m_{\omega }=-J_{\omega }}^{J_{\omega }}D_{m_{X}m_{b_{1}}}^{L_{X}*}(\theta _{X},\phi _{X},0)D_{m_{b_{1}}m_{\omega }}^{J_{b_{1}}*}(\theta _{b_{1}},\phi _{b_{1}},0)$	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 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 (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_{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^-}} }
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 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 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 \omega (J_\omega^{PC}=1^{--})} breakup angular distributions, currently modeled as 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 L_{\omega\rightarrow\pi^0+\rho}=0; L_{\rho\rightarrow\pi^++\pi^-}=1=L_{\pi^+\pi^-}}
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} 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.
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_\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: 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 \pi^0 \oplus (\pi^+ \oplus \pi^-) \rightarrow \omega}