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Amplitudes for the Exotic b1π Decay (view source)
Revision as of 00:00, 20 August 2011
, 00:00, 20 August 2011no edit summary
:<math>
:<math>
\langle J M \epsilon|\mathbb{R}^{-1} V \mathbb{R}|
\langle J M \epsilon|\mathbb{R}^{-1} V \mathbb{R}|
\epsilon_\gamma ; J_R \lambda_R \epsilon_R ; t, s; \Omega_0 \rangle =
\epsilon_\gamma ; \lambda_R \epsilon_R ; t, s; \Omega_0 \rangle =
\epsilon \epsilon_\gamma \epsilon_R \langle J M \epsilon|V|
\epsilon \epsilon_\gamma \epsilon_R \langle J M \epsilon|V|
\epsilon_\gamma ; J_R \lambda_R \epsilon_R ; \Omega_0 \rangle
\epsilon_\gamma ; \lambda_R \epsilon_R ; \Omega_0 \rangle
</math>
</math>
:<math>
:<math>
\langle J M \epsilon|V|
\langle J M \epsilon|V|
\epsilon_\gamma ; J_R \lambda_R \epsilon_R ; \Omega_0 \rangle
\epsilon_\gamma ; \lambda_R \epsilon_R ; \Omega_0 \rangle
= v\left(J,m,\epsilon;\epsilon_\gamma;J_R,\lambda_R,\epsilon_R \right)
= v\left(J,m,\epsilon;\epsilon_\gamma; \lambda_R,\epsilon_R \right)
</math>
</math>
=\sum_{R,\lambda_R,\epsilon_R;\lambda_{b_1},\lambda_\omega,\lambda_\rho}
=\sum_{R,\lambda_R,\epsilon_R;\lambda_{b_1},\lambda_\omega,\lambda_\rho}
\langle \mathbf{q}_\pi 0 0; \mathbf{q}_\rho \lambda_\rho 0; \mathbf{q}_\omega \lambda_\omega 0; \mathbf{q}_{b1}\lambda_{b1} 0
\langle \mathbf{q}_\pi 0 0; \mathbf{q}_\rho \lambda_\rho 0; \mathbf{q}_\omega \lambda_\omega 0; \mathbf{q}_{b1}\lambda_{b1} 0
| UV | \epsilon_\gamma; J_R \lambda_R \epsilon_R; \Omega_0\rangle
| UV | \epsilon_\gamma; \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 \lambda_R \epsilon_R; \Omega_0;\mathbf{p_f}, \epsilon_f | W | \mathbf{p_i}, \epsilon_i\rangle
</math>
</math>
:::::::::<math>\times
:::::::::<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; \Omega_0\rangle
\epsilon_\gamma; \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 \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 parameterized them into a pair of unknown functions ''v(s,t)'' and ''w(s,t)''.
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)''.
v^{X,M_X,\epsilon_X}_{\lambda_R,\epsilon_R}(s,t) =
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; \lambda_R \epsilon_R; \Omega_0\rangle
</math>
</math>
:<math>\displaystyle
:<math>\displaystyle
w_{\epsilon_R;\epsilon_i} =
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 \lambda_R \epsilon_R; \Omega_0;\mathbf{p_f}, \epsilon_f | W | \mathbf{p_i}, \epsilon_i\rangle
</math>
</math>
T_{\pm 1}=\sum_{R,\lambda_R;\lambda_{b_1},\lambda_\omega,\lambda_\rho}
T_{\pm 1}=\sum_{R,\lambda_R;\lambda_{b_1},\lambda_\omega,\lambda_\rho}
\langle \mathbf{q}_\pi 0 0; \mathbf{q}_\rho \lambda_\rho 0; \mathbf{q}_\omega \lambda_\omega 0; \mathbf{q}_{b1}\lambda_{b1} 0
\langle \mathbf{q}_\pi 0 0; \mathbf{q}_\rho \lambda_\rho 0; \mathbf{q}_\omega \lambda_\omega 0; \mathbf{q}_{b1}\lambda_{b1} 0
| U | (1\; \pm 1)_{\mathrm{lab}}; J_R \lambda_R \epsilon_R;s,t \rangle
| U | (1\; \pm 1)_{\mathrm{lab}}; \lambda_R \epsilon_R;s,t \rangle
</math>
</math>