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Amplitudes for the Exotic b1π Decay (view source)
Revision as of 15:29, 16 August 2011
, 15:29, 16 August 2011→Decay of t-channel resonance X to b1π
\langle \Omega_{b_1} \lambda_{b_1} 0| U_X | J_X M_X \rangle
\langle \Omega_{b_1} \lambda_{b_1} 0| U_X | J_X M_X \rangle
=\sum_{L_X}
=\sum_{L_X}
\left[ \sqrt{\frac{2J_X+1}{4\pi}} D_{m_X \lambda_{b_1}}^{J_X *}(\Omega_{b_1}) \right]
\left[ \sqrt{\frac{2J_X+1}{4\pi}} D_{M_X \lambda_{b_1}}^{J_X *}(\Omega_{b_1}) \right]
\left[ \sqrt{\frac{2L_X+1}{2J_X+1}}
\left[ \sqrt{\frac{2L_X+1}{2J_X+1}}
\left(\begin{array}{cc|c}
\left(\begin{array}{cc|c}
\langle \Omega_\omega \lambda_\omega 0| U_{b_1} | 1 , M_{b_1}=\lambda_{b_1} \rangle
\langle \Omega_\omega \lambda_\omega 0| U_{b_1} | 1 , M_{b_1}=\lambda_{b_1} \rangle
=\sum_{L_{b_1}}
=\sum_{L_{b_1}}
\left[ \sqrt{\frac{2J_{b_1}+1}{4\pi}} D_{m_{b_1} \lambda_\omega}^{1 *}(\Omega_\omega) \right]
\left[ \sqrt{\frac{2J_{b_1}+1}{4\pi}} D_{M_{b_1} \lambda_\omega}^{1 *}(\Omega_\omega) \right]
\left[ \sqrt{\frac{2L_{b_1}+1}{2J_{b_1}+1}}
\left[ \sqrt{\frac{2L_{b_1}+1}{2J_{b_1}+1}}
\left(\begin{array}{cc|c}
\left(\begin{array}{cc|c}
\langle \Omega_\rho \lambda_\rho 0| U_\omega | 1 , M_\omega=\lambda_\omega \rangle
\langle \Omega_\rho \lambda_\rho 0| U_\omega | 1 , M_\omega=\lambda_\omega \rangle
=\sum_{L_\omega J_\rho}
=\sum_{L_\omega J_\rho}
\left[ \sqrt{\frac{2J_\omega+1}{4\pi}} D_{m_\omega \lambda_\rho}^{1 *}(\Omega_\rho) \right]
\left[ \sqrt{\frac{2J_\omega+1}{4\pi}} D_{M_\omega \lambda_\rho}^{1 *}(\Omega_\rho) \right]
\left[ \sqrt{\frac{2L_\omega+1}{2J_\omega+1}}
\left[ \sqrt{\frac{2L_\omega+1}{2J_\omega+1}}
\left(\begin{array}{cc|c}
\left(\begin{array}{cc|c}
\langle \Omega_{\pi} 0 0 | U_\rho | J_\rho , M_\rho=\lambda_\rho \rangle
\langle \Omega_{\pi} 0 0 | U_\rho | J_\rho , M_\rho=\lambda_\rho \rangle
=\sum_{L_\rho}
=\sum_{L_\rho}
\left[ \sqrt{\frac{2J_\rho+1}{4\pi}} D_{m_\rho 0}^{J_\rho *}(\Omega_{\pi}) \right]
\left[ \sqrt{\frac{2J_\rho+1}{4\pi}} D_{M_\rho 0}^{J_\rho *}(\Omega_{\pi}) \right]
\left[ \sqrt{\frac{2L_\rho+1}{2J_\rho+1}}
\left[ \sqrt{\frac{2L_\rho+1}{2J_\rho+1}}
\left(\begin{array}{cc|c}
\left(\begin{array}{cc|c}
</math>
</math>
:::::::::<math>
:::::::::<math>
=Y_{m_\rho}^{J_\rho}(\Omega_{\pi})
=Y_{M_\rho}^{J_\rho}(\Omega_{\pi})
f_{J_\rho\,0}^{J_\rho}
f_{J_\rho\,0}^{J_\rho}
</math>
</math>