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Line 242:
Line 242: − + − f^{X,M_X,\epsilon_X}_{R,\lambda_R,\epsilon_R;\epsilon_i,\epsilon_f}(s,t) = + − :::::::::<math>\times+ +
Amplitudes for the Exotic b1π Decay (view source)
Revision as of 23:55, 19 August 2011
, 23:55, 19 August 2011→Assembly of the full amplitude
\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>
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.
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 parameterized them into a pair of unknown functions ''v(s,t)'' and ''w(s,t)''.
:<math>\displaystyle
:<math>\displaystyle
v^{X,M_X,\epsilon_X}_{\lambda_R,\epsilon_R}(s,t) =
\langle J_X M_X \epsilon_X | V |
\langle J_X M_X \epsilon_X | V |
\epsilon_\gamma; J_R \lambda_R \epsilon_R; \Omega_0\rangle
\epsilon_\gamma; J_R \lambda_R \epsilon_R; \Omega_0\rangle
</math>
</math>
:<math>\displaystyle
w_{\epsilon_R;\epsilon_i} =
\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>