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Amplitudes for the Exotic b1π Decay (view source)
Revision as of 19:45, 15 August 2011
, 19:45, 15 August 2011→Assembly of the full amplitude
\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>
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}=