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Line 149: − + − \sqrt{\frac{2L_\rho+1}{4\pi}}+ − − d_{L_\rho} + + +
Amplitudes for the Exotic b1π Decay (view source)
Revision as of 22:22, 12 August 2011
, 22:22, 12 August 2011→Decay of t-channel resonance X
<math>
<math>
\langle \Omega_X 0 \lambda_{b_1} | U_X | J_X m_X \rangle
\langle \Omega_{b_1} 0 \lambda_{b_1} | 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_X,0) \right]
\left[ \sqrt{\frac{2J_X+1}{4\pi}} D_{m_X \lambda_{b_1}}^{J_X *}(\Omega_X,0) \right]
<math>
<math>
\langle \Omega_{b_1} 0 \lambda_\omega | U_{b_1} | 1 , m_{b_1}=\lambda_{b_1} \rangle
\langle \Omega_\omega 0 \lambda_\omega | 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_{b_1} \lambda_\omega}^{1 *}(\Omega_{b_1},0) \right]
\left[ \sqrt{\frac{2J_{b_1}+1}{4\pi}} D_{m_{b_1}=\lambda_{b_1} \lambda_\omega}^{1 *}(\Omega_{b_1},0) \right]
\end{array}\right)
\end{array}\right)
\right]
\right]
b_{L_{b_1}}
b_{L_{b_1}}^1
</math>
</math>
<math>
<math>
\langle \Omega_\omega 0 \lambda_\rho | U_\omega | 1 , m_\omega=\lambda_\omega \rangle
\langle \Omega_\rho 0 \lambda_\rho | 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_\omega \lambda_\rho}^{1 *}(\Omega_\omega,0) \right]
\left[ \sqrt{\frac{2J_\omega+1}{4\pi}} D_{m_\omega=\lambda_\omega \lambda_\rho}^{1 *}(\Omega_\omega,0) \right]
\end{array}\right)
\end{array}\right)
\right]
\right]
c_{L_\omega J_\rho}
c_{L_\omega J_\rho}^1
</math>
</math>
<math>
<math>
\langle \Omega_\rho 0 \lambda_\rho | U_\rho | J_\rho , m_\rho=\lambda_\rho \rangle
\langle \Omega_{\pi^+} 0 \lambda_\rho | 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_\rho,0) \right]
\left[ \sqrt{\frac{2J_\rho+1}{4\pi}} D_{m_\rho 0}^{J_\rho *}(\Omega_\rho,0) \right]
\right]
\right]
d_{L_\rho}
d_{L_\rho}
=\sum_{L_\rho}
=Y_{m_\rho}^{J_\rho *}(\Omega_\rho)
f_{J_\rho}^{J_\rho}
Y_{m_\rho}^{J_\rho *}(\Omega_\rho)
</math>
</math>
* change subscripts on \Omega (e.g. X->b1)
* don't write m=lambda redundantly
* be consistent about indexing of known spins