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| \langle J_R \lambda_R \epsilon_R; \mathbf{p_f}, \lambda_f | W | \mathbf{p_i}, \lambda_i\rangle | | \langle J_R \lambda_R \epsilon_R; \mathbf{p_f}, \lambda_f | W | \mathbf{p_i}, \lambda_i\rangle |
| </math> | | </math> |
− | | + | The aggregate decay matrix element can be further broken up into a product of individual decay amplitudes, |
| :<math> | | :<math> |
− | \sum_{L_X \epsilon_R} A_{L_X \epsilon_\gamma \epsilon_R}^{J_X}= | + | \langle \Omega_\pi 0 0; \Omega_\rho \lambda_\rho 0; \Omega_\omega \lambda_\omega 0; \Omega_{b1} \lambda_{b1} 0 |
− | \sum_{\lambda_R,\lambda_{b_1},\lambda_\omega,\lambda_\rho} | + | | U | \epsilon_\gamma; J_R \lambda_R \epsilon_R \rangle |
| + | </math> |
| + | ::<math>=\sum_{\lambda_R,\lambda_{b_1},\lambda_\omega,\lambda_\rho} |
| \langle \Omega_X \lambda_{b_1} 0| U_X | | | \langle \Omega_X \lambda_{b_1} 0| U_X | |
| \left\{|J_X m_X \rangle + \epsilon_\gamma \epsilon_R P_X (-1)^{J_X-m_X} |J_X\;-m_X \rangle \right\} | | \left\{|J_X m_X \rangle + \epsilon_\gamma \epsilon_R P_X (-1)^{J_X-m_X} |J_X\;-m_X \rangle \right\} |
| </math> | | </math> |
− | ::<math> | + | :::<math> |
− | \langle \Omega_{b_1} \lambda_\omega 0| U_{b_1} | 1 , m_{b_1}=\lambda_{b_1} \rangle | + | \langle \Omega_{b_1} \lambda_\omega 0| U_{b_1} | 1 , \lambda_{b_1} \rangle |
− | \langle \Omega_\omega \lambda_\rho 0| U_\omega | 1 , m_\omega=\lambda_\omega \rangle | + | \langle \Omega_\omega \lambda_\rho 0| U_\omega | 1 , \lambda_\omega \rangle |
− | \langle \Omega_\rho 0 0 | U_\rho | J_\rho , m_\rho=\lambda_\rho \rangle C_\rho(L_\rho) | + | \langle \Omega_\rho 0 0 | U_\rho | J_\rho , \lambda_\rho \rangle C_\rho(L_\rho) |
| </math> | | </math> |
| ::<math> | | ::<math> |
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| | | |
| where f is the polarization fraction varying from 1, 100% x-polarized, to 0, unpolarized. | | where f is the polarization fraction varying from 1, 100% x-polarized, to 0, unpolarized. |
| + | \sum_{L_X \epsilon_R} A_{L_X \epsilon_\gamma \epsilon_R}^{J_X}= |