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Amplitudes for the Exotic b1π Decay (view source)
Revision as of 16:02, 12 July 2011
, 16:02, 12 July 2011no edit summary
<math>
<table>
<tr>
<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_{b1}}^{J_{b1}}
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_{b1} n_{b1}}^{J_{b1}*}(\theta_{b1},\phi_{b1},0)
</math></td>
<td>
angular distributions two-body X and b1 decays
</td>
</tr>
<tr>
<td><math>
\left[
\left[
(-)^{J_X+1+\epsilon} e^{2i\alpha}
(-)^{J_X+1+\epsilon} e^{2i\alpha}
\left(\begin{array}{cc|c}
\left(\begin{array}{cc|c}
J_{b1} & L_X & J_X \\
J_{b1} & L_X & J_X \\
m_{b1} & m_X & -1
m_{b1} & m_X & +1
\end{array}\right)
\end{array}\right)
\right]
\right]
</math></td>
<td>
resonance helicity sum
</td>
</tr>
<tr>
<td><math>
\left(\frac{1+(-)^\epsilon \eta}{4}\right)
\left(\frac{1+(-)^\epsilon \eta}{4}\right)
</math></td>
<td>
polarization term: ε=0(1) for x (y) polarization; η is the polarization fraction
</td>
</tr>
<tr>
<td><math>
k^{L_X} q^{J_{b1}}
k^{L_X} q^{J_{b1}}
</math></td>
<td>
k, q are breakup momenta for the resonance and isobar, respectively
</td>
</tr>
<tr>
<td><math>
\left(\begin{array}{cc|c}
\left(\begin{array}{cc|c}
I_{b1} & 1 & I_X \\
I_{b1} & 1 & 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)
</math></td>
<td>
Clebsch-Gordan coefficients for isospin sum <math>b1 \oplus \pi^- \rightarrow X</math>
</td>
</tr>
<tr>
<td><math>
\sum\limits_{L_{b1}=0}^{2} \sum\limits_{m_{L_{b1}}=-L_{b1}}^{L_{b1}}
\sum\limits_{L_{b1}=0}^{2} \sum\limits_{m_{L_{b1}}=-L_{b1}}^{L_{b1}}
D_{m_\omega \lambda_\rho}^{J_\omega *}(\theta_\omega,\phi_\omega,0)
Y_{m_\rho}^{s_\rho}(\theta_\rho,\phi_\rho)
</math></td>
<td>
two-stage ω breakup angular distributions
</td>
</tr>
<tr>
<td><math>
\left(\begin{array}{cc|c}
\left(\begin{array}{cc|c}
s_\omega & L_{b1} & J_{b1} \\
s_\omega & L_{b1} & J_{b1} \\
0 & m_{L_{b1}} & m_{b1}
0 & m_{L_{b1}} & m_{b1}
\end{array}\right)
\end{array}\right)
\left(\begin{array}{cc|c}
\left(\begin{array}{cc|c}
1 & s_\rho & J_\omega \\
1 & s_\rho & J_\omega \\
0 & \lambda_\rho & m_\omega
0 & \lambda_\rho & m_\omega
\end{array}\right)
\end{array}\right)
</math></td>
<td>
angular momentum sum Clebsch-Gordan coefficients for b1 and ω decays
</td>
</tr>
<tr>
<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 & -1 & I_{z\rho}
+1 & -1 & I_{z\rho}
\end{array}\right)
\end{array}\right)
</math></td>
<td>
Clebsch-Gordan coefficients for isospin sums: <math>\pi^0 \oplus (\pi^+ \oplus \pi^-) \rightarrow \omega</math>
</td>
</math>
</tr>
</table>