Difference between revisions of "Amplitudes for the Exotic b1π Decay"
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<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... | ||
| Line 7: | Line 10: | ||
<tr> | <tr> | ||
<td><math> | <td><math> | ||
| − | \sum\limits_{m_X=-L_X}^{L_X} \sum\limits_{m_{b1}=-J_{ | + | \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_{ | + | D_{m_{b_1} m_\omega}^{J_{b_1}*}(\theta_{b_1},\phi_{b_1},0) |
</math></td> | </math></td> | ||
<td> | <td> | ||
| Line 20: | Line 25: | ||
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_{ | + | J_{b_1} & L_X & J_X \\ |
| − | m_{ | + | m_{b_1} & m_X & -1 |
\end{array}\right) | \end{array}\right) | ||
+ | + | ||
\left(\begin{array}{cc|c} | \left(\begin{array}{cc|c} | ||
| − | J_{ | + | J_{b_1} & L_X & J_X \\ |
| − | m_{ | + | 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; < | + | resonance helicity sum: ε=0 (1) for x (y) polarization; <math>P_X</math> is the parity of the resonance |
</td> | </td> | ||
</tr> | </tr> | ||
| Line 44: | Line 49: | ||
<tr> | <tr> | ||
<td><math> | <td><math> | ||
| − | k^{L_X} q^{ | + | k^{L_X} q^{L_{b_1}} |
</math></td> | </math></td> | ||
<td> | <td> | ||
| Line 53: | Line 58: | ||
<td><math> | <td><math> | ||
\left(\begin{array}{cc|c} | \left(\begin{array}{cc|c} | ||
| − | I_{ | + | 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) | ||
| Line 63: | Line 68: | ||
<tr> | <tr> | ||
<td><math> | <td><math> | ||
| − | \sum\limits_{L_{ | + | \sum\limits_{L_{b_1}=0}^{2} |
| − | \sum\limits_{m_{L_{ | + | \sum\limits_{m_{L_{b_1}}=-L_{b_1}}^{L_{b_1}} |
| − | + | \sum\limits_{\lambda_\rho=-s_\rho}^{s_\rho} | |
| − | \sum\limits_{\lambda_\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) | ||
| Line 72: | Line 76: | ||
<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 | + | currently modeled as <math>L_{\omega\rightarrow\pi^0+\rho}=0; L_{\rho\rightarrow\pi^++\pi^-}=1=s_\rho</math> |
</td> | </td> | ||
</tr> | </tr> | ||
| Line 78: | Line 82: | ||
<td><math> | <td><math> | ||
\left(\begin{array}{cc|c} | \left(\begin{array}{cc|c} | ||
| − | J_\omega & L_{ | + | J_\omega & L_{b_1} & J_{b_1} \\ |
| − | m_\omega & m_{L_{ | + | 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} | ||
| Line 92: | Line 96: | ||
<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 \\ | ||
Revision as of 00:19, 13 July 2011
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_{}^{J_X L_X P_X}= } |
defining an amplitude... |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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) D_{m_{b_1} m_\omega}^{J_{b_1}*}(\theta_{b_1},\phi_{b_1},0) } |
angular distributions two-body X and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b_1 (J_{b_1}^{PC}=1^{+-})} decays |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left[ P_X(-)^{J_X+1+\epsilon} e^{2i\alpha} \left(\begin{array}{cc|c} J_{b_1} & L_X & J_X \\ m_{b_1} & m_X & -1 \end{array}\right) + \left(\begin{array}{cc|c} J_{b_1} & L_X & J_X \\ m_{b_1} & m_X & +1 \end{array}\right) \right] } |
resonance helicity sum: ε=0 (1) for x (y) polarization; Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_X} is the parity of the resonance |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(\frac{1+(-)^\epsilon \eta}{4}\right) } |
polarization term: η is the polarization fraction |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k^{L_X} q^{L_{b_1}} } |
k, q are breakup momenta for the resonance and isobar, respectively |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(\begin{array}{cc|c} I_{b_1} & I_\pi & I_X \\ I_{z\pi^+} & I_{z\pi^-} & I_{z\pi^+}+I_{z\pi^-} \end{array}\right) } |
Clebsch-Gordan coefficients for isospin sum Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b_1 \oplus \pi^- \rightarrow X} |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum\limits_{L_{b_1}=0}^{2} \sum\limits_{m_{L_{b_1}}=-L_{b_1}}^{L_{b_1}} \sum\limits_{\lambda_\rho=-s_\rho}^{s_\rho} D_{m_\omega \lambda_\rho}^{J_\omega *}(\theta_\omega,\phi_\omega,0) Y_{\lambda_\rho}^{s_\rho}(\theta_\rho,\phi_\rho) } |
two-stage Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega (J_\omega^{PC}=1^{--})} breakup angular distributions, currently modeled as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_{\omega\rightarrow\pi^0+\rho}=0; L_{\rho\rightarrow\pi^++\pi^-}=1=s_\rho} |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(\begin{array}{cc|c} J_\omega & L_{b_1} & J_{b_1} \\ m_\omega & m_{L_{b_1}} & m_{b_1} \end{array}\right) \left(\begin{array}{cc|c} 1 & s_\rho & J_\omega \\ 0 & \lambda_\rho & m_\omega \end{array}\right) } |
angular momentum sum Clebsch-Gordan coefficients for b1 and ω decays. |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum\limits_{I_\rho=0}^{1} \sum\limits_{I_{z\rho}=-I_\rho}^{I_\rho} \left(\begin{array}{cc|c} 1 & I_\rho & 0 \\ 0 & I_{z\rho} & 0 \end{array}\right) \left(\begin{array}{cc|c} I_{\pi} & I_{\pi} & I_\rho \\ +1 & -1 & I_{z\rho} \end{array}\right) } |
Clebsch-Gordan coefficients for isospin sums: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi^0 \oplus (\pi^+ \oplus \pi^-) \rightarrow \omega} |