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Line 230: − + + + + + + − + + + + + + + + − + − + − \langle \mathbf{q}_\pi 0 0 | U_\rho | J_\rho , \lambda_\rho \rangle − + −
Amplitudes for the Exotic b1π Decay (view source)
Revision as of 22:53, 19 August 2011
, 22:53, 19 August 2011→Assembly of the full amplitude
X,M_X,\epsilon_X;\epsilon_i,\epsilon_f
X,M_X,\epsilon_X;\epsilon_i,\epsilon_f
\end{array}}
\end{array}}
\langle \mathbf{q}_{b1} \lambda_{b_1} 0| U_X | J_X M_X \epsilon_X\rangle
\langle \mathbf{q}_{b1} \lambda_{b_1} 0| U_X | J_X M_X \epsilon_X\rangle \langle \mathbf{q}_\omega \lambda_\omega 0| U_{b_1} | 1 , \lambda_{b_1} \rangle
</math>
:::::::::<math>\times
\langle \mathbf{q}_\rho \lambda_\rho 0| U_\omega | 1 , \lambda_\omega \rangle \langle \mathbf{q}_\pi 0 0 | U_\rho | J_\rho , \lambda_\rho \rangle
</math>
:::::::::<math>\times
\langle J_X M_X \epsilon_X | V |
\langle J_X M_X \epsilon_X | V |
\epsilon_\gamma; J_R \lambda_R \epsilon_R; s,t \rangle
\epsilon_\gamma; J_R \lambda_R \epsilon_R; \Omega_0\rangle
</math>
:::::::::<math>\times
\langle J_R \lambda_R \epsilon_R; \Omega_0;\mathbf{p_f}, \epsilon_f | W | \mathbf{p_i}, \epsilon_i\rangle
</math>
Parity conservation requires that <math>\epsilon_X=\epsilon_\gamma \epsilon_R</math> and <math>\epsilon_R=\epsilon_i\epsilon_f</math>. The last two matrix elements in the expression above for <math>T_{fi}</math> not known ''a priori'', so we combine them into a single factor that expresses the net amplitude for producing resonance ''X'' in the given state from the given initial state.
:<math>\displaystyle
f_X(M_X,\epsilon_X,R,\lambda_R,\epsilon_R,\epsilon_i,\epsilon_f) =
</math>
</math>
:::::::::<math>\times
:::::::::<math>\times
\langle \mathbf{q}_\omega \lambda_\omega 0| U_{b_1} | 1 , \lambda_{b_1} \rangle
\langle J_X M_X \epsilon_X | V |
\langle \mathbf{q}_\rho \lambda_\rho 0| U_\omega | 1 , \lambda_\omega \rangle
\epsilon_\gamma; J_R \lambda_R \epsilon_R; \Omega_0\rangle
</math>
</math>
<math>\times
:::::::::<math>\times
\langle J_R \lambda_R \epsilon_R; \Omega_0;\mathbf{p_f}, \epsilon_f | W | \mathbf{p_i}, \epsilon_i\rangle
\langle J_R \lambda_R \epsilon_R; \Omega_0;\mathbf{p_f}, \epsilon_f | W | \mathbf{p_i}, \epsilon_i\rangle
</math>
</math>
=== Proton states and individual decay amplitudes ===
=== Proton states and individual decay amplitudes ===