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Amplitudes for the Exotic b1π Decay (view source)
Revision as of 00:19, 13 July 2011
, 00:19, 13 July 2011no edit summary
<table>
<table>
A_{}^{J_X L_X P_X}
<tr>
<td><math>
A_{}^{J_X L_X P_X}=
</math></td>
<td>
<td>
defining an amplitude...
defining an amplitude...
<tr>
<tr>
<td><math>
<td><math>
\sum\limits_{m_X=-L_X}^{L_X} \sum\limits_{m_{b1}=-J_{b1}}^{J_{b1}}
\sum\limits_{m_X=-L_X}^{L_X}
\sum\limits_{m_{b1}=-J_{b_1}}^{J_{b_1}}
\sum\limits_{m_\omega=-J_\omega}^{J_\omega}
Y_{m_X}^{L_X}(\theta_X,\phi_X)
Y_{m_X}^{L_X}(\theta_X,\phi_X)
D_{m_{b1} n_{b1}}^{J_{b1}*}(\theta_{b1},\phi_{b1},0)
D_{m_{b_1} m_\omega}^{J_{b_1}*}(\theta_{b_1},\phi_{b_1},0)
</math></td>
</math></td>
<td>
<td>
P_X(-)^{J_X+1+\epsilon} e^{2i\alpha}
P_X(-)^{J_X+1+\epsilon} e^{2i\alpha}
\left(\begin{array}{cc|c}
\left(\begin{array}{cc|c}
J_{b1} & L_X & J_X \\
J_{b_1} & L_X & J_X \\
m_{b1} & m_X & -1
m_{b_1} & m_X & -1
\end{array}\right)
\end{array}\right)
+
+
\left(\begin{array}{cc|c}
\left(\begin{array}{cc|c}
J_{b1} & L_X & J_X \\
J_{b_1} & L_X & J_X \\
m_{b1} & m_X & +1
m_{b_1} & m_X & +1
\end{array}\right)
\end{array}\right)
\right]
\right]
</math></td>
</math></td>
<td>
<td>
resonance helicity sum: ε=0 (1) for x (y) polarization; </math>P_X</math> is the parity of the resonance
resonance helicity sum: ε=0 (1) for x (y) polarization; <math>P_X</math> is the parity of the resonance
</td>
</td>
</tr>
</tr>
<tr>
<tr>
<td><math>
<td><math>
k^{L_X} q^{J_{b1}}
k^{L_X} q^{L_{b_1}}
</math></td>
</math></td>
<td>
<td>
<td><math>
<td><math>
\left(\begin{array}{cc|c}
\left(\begin{array}{cc|c}
I_{b1} & I_\pi & I_X \\
I_{b_1} & I_\pi & I_X \\
I_{z\pi^+} & I_{z\pi^-} & I_{z\pi^+}+I_{z\pi^-}
I_{z\pi^+} & I_{z\pi^-} & I_{z\pi^+}+I_{z\pi^-}
\end{array}\right)
\end{array}\right)
<tr>
<tr>
<td><math>
<td><math>
\sum\limits_{L_{b1}=0}^{2}
\sum\limits_{L_{b_1}=0}^{2}
\sum\limits_{m_{L_{b1}}=-L_{b1}}^{L_{b1}}
\sum\limits_{m_{L_{b_1}}=-L_{b_1}}^{L_{b_1}}
\sum\limits_{\lambda_\rho=-s_\rho}^{s_\rho}
\sum\limits_{\lambda_\rho}=-s_\rho}^{s_\rho}
D_{m_\omega \lambda_\rho}^{J_\omega *}(\theta_\omega,\phi_\omega,0)
D_{m_\omega \lambda_\rho}^{J_\omega *}(\theta_\omega,\phi_\omega,0)
Y_{\lambda_\rho}^{s_\rho}(\theta_\rho,\phi_\rho)
Y_{\lambda_\rho}^{s_\rho}(\theta_\rho,\phi_\rho)
<td>
<td>
two-stage <math>\omega (J_\omega^{PC}=1^{--})</math> breakup angular distributions,
two-stage <math>\omega (J_\omega^{PC}=1^{--})</math> breakup angular distributions,
currently modeled as <math>L_{\omega\rightarrow\pi^0-\rho}=0; L_{\rho\rightarrow\pi^++\pi^-}=1=s_\rho</math>
currently modeled as <math>L_{\omega\rightarrow\pi^0+\rho}=0; L_{\rho\rightarrow\pi^++\pi^-}=1=s_\rho</math>
</td>
</td>
</tr>
</tr>
<td><math>
<td><math>
\left(\begin{array}{cc|c}
\left(\begin{array}{cc|c}
J_\omega & L_{b1} & J_{b1} \\
J_\omega & L_{b_1} & J_{b_1} \\
m_\omega & m_{L_{b1}} & m_{b1}
m_\omega & m_{L_{b_1}} & m_{b_1}
\end{array}\right)
\end{array}\right)
\left(\begin{array}{cc|c}
\left(\begin{array}{cc|c}
<tr>
<tr>
<td><math>
<td><math>
\sum\limits_{I_\rho=0}^{1} \sum\limits_{I_{z\rho}=-I_\rho}^{I_\rho}
\sum\limits_{I_\rho=0}^{1}
\sum\limits_{I_{z\rho}=-I_\rho}^{I_\rho}
\left(\begin{array}{cc|c}
\left(\begin{array}{cc|c}
1 & I_\rho & 0 \\
1 & I_\rho & 0 \\