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...
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
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_{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; 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 P_X} 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^{L_{b_1}} }	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_{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}