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Line 153: − * don't write m=lambda redundantly − * be consistent about indexing of known spins − + − + − A^{J_X}=\sum_{\lambda_{b_1},\lambda_\omega,\lambda_\rho} + − + − + − + − + + + + + + + +
Amplitudes for the Exotic b1π Decay (view source)
Revision as of 00:23, 13 August 2011
, 00:23, 13 August 2011no edit summary
:<math>
:<math>
=\sum_{L,S}
=\sum_{L,S}
\left[ \sqrt{\frac{2J+1}{4\pi}} D_{m \lambda}^{J *}(\Omega,0) \right]
\left[ \sqrt{\frac{2J+1}{4\pi}} D_{m \lambda}^{J *}(\Omega) \right]
\left[ \sqrt{\frac{2L+1}{2J+1}}
\left[ \sqrt{\frac{2L+1}{2J+1}}
\left(\begin{array}{cc|c}
\left(\begin{array}{cc|c}
a_{L S}^{J}
a_{L S}^{J}
</math>
</math>
where <math>\lambda=\lambda_1-\lambda_2</math>,and the double-stacked symbols are Clebsh-Gordon coefficients. The product of CG coefficients in the second brackets on the right-hand side represent the overlap between the basis vectors in the helicity and ''L,S'' basis, which turns out to be independent of ''m'', as required by rotational invariance. This overlap integral is somewhat lengthy to calculate, but the result turns out to be fairly simple, as shown above. This expression holds regardless of what axis is used to define the quantization direction for m, but whatever choice is made must serve as the z-axis of the reference frame in which the plane wave direction <math>\Omega</math> is defined.
where <math>\lambda=\lambda_1-\lambda_2</math>, <math>\Omega=(\phi,\theta,0)</math> and the double-stacked symbols are Clebsh-Gordon coefficients. The product of CG coefficients in the second brackets on the right-hand side represent the overlap between the basis vectors in the helicity and ''L,S'' basis, which turns out to be independent of ''m'', as required by rotational invariance. This overlap integral is somewhat lengthy to calculate, but the result turns out to be fairly simple, as shown above. This expression holds regardless of what axis is used to define the quantization direction for m, but whatever choice is made must serve as the z-axis of the reference frame in which the plane wave direction <math>\Omega</math> is defined.
== Isospin Projections ==
== Isospin Projections ==
\langle \Omega_{b_1} 0 \lambda_{b_1} | U_X | J_X m_X \rangle
\langle \Omega_{b_1} 0 \lambda_{b_1} | U_X | J_X m_X \rangle
=\sum_{L_X}
=\sum_{L_X}
\left[ \sqrt{\frac{2J_X+1}{4\pi}} D_{m_X \lambda_{b_1}}^{J_X *}(\Omega_{b_1},0) \right]
\left[ \sqrt{\frac{2J_X+1}{4\pi}} D_{m_X \lambda_{b_1}}^{J_X *}(\Omega_{b_1}) \right]
\left[ \sqrt{\frac{2L_X+1}{2J_X+1}}
\left[ \sqrt{\frac{2L_X+1}{2J_X+1}}
\left(\begin{array}{cc|c}
\left(\begin{array}{cc|c}
\langle \Omega_\omega 0 \lambda_\omega | U_{b_1} | 1 , m_{b_1}=\lambda_{b_1} \rangle
\langle \Omega_\omega 0 \lambda_\omega | U_{b_1} | 1 , m_{b_1}=\lambda_{b_1} \rangle
=\sum_{L_{b_1}}
=\sum_{L_{b_1}}
\left[ \sqrt{\frac{2J_{b_1}+1}{4\pi}} D_{m_{b_1} \lambda_\omega}^{1 *}(\Omega_\omega,0) \right]
\left[ \sqrt{\frac{2J_{b_1}+1}{4\pi}} D_{m_{b_1} \lambda_\omega}^{1 *}(\Omega_\omega) \right]
\left[ \sqrt{\frac{2L_{b_1}+1}{2J_{b_1}+1}}
\left[ \sqrt{\frac{2L_{b_1}+1}{2J_{b_1}+1}}
\left(\begin{array}{cc|c}
\left(\begin{array}{cc|c}
\langle \Omega_\rho 0 \lambda_\rho | U_\omega | 1 , m_\omega=\lambda_\omega \rangle
\langle \Omega_\rho 0 \lambda_\rho | U_\omega | 1 , m_\omega=\lambda_\omega \rangle
=\sum_{L_\omega J_\rho}
=\sum_{L_\omega J_\rho}
\left[ \sqrt{\frac{2J_\omega+1}{4\pi}} D_{m_\omega \lambda_\rho}^{1 *}(\Omega_\rho,0) \right]
\left[ \sqrt{\frac{2J_\omega+1}{4\pi}} D_{m_\omega \lambda_\rho}^{1 *}(\Omega_\rho) \right]
\left[ \sqrt{\frac{2L_\omega+1}{2J_\omega+1}}
\left[ \sqrt{\frac{2L_\omega+1}{2J_\omega+1}}
\left(\begin{array}{cc|c}
\left(\begin{array}{cc|c}
\langle \Omega_{\pi^+} 0 \lambda_\rho | U_\rho | J_\rho , m_\rho=\lambda_\rho \rangle
\langle \Omega_{\pi^+} 0 \lambda_\rho | U_\rho | J_\rho , m_\rho=\lambda_\rho \rangle
=\sum_{L_\rho}
=\sum_{L_\rho}
\left[ \sqrt{\frac{2J_\rho+1}{4\pi}} D_{m_\rho 0}^{J_\rho *}(\Omega_{\pi^+},0) \right]
\left[ \sqrt{\frac{2J_\rho+1}{4\pi}} D_{m_\rho 0}^{J_\rho *}(\Omega_{\pi^+}) \right]
\left[ \sqrt{\frac{2L_\rho+1}{2J_\rho+1}}
\left[ \sqrt{\frac{2L_\rho+1}{2J_\rho+1}}
\left(\begin{array}{cc|c}
\left(\begin{array}{cc|c}
</math>
</math>
* apply reflectivity on the original J_X,m_X state
* apply reflectivity on the original J_X,m_X state
:<math>
<math>
\sum_{L_X} A_{L_X}^{J_X}=
\sum_{\lambda_{b_1},\lambda_\omega,\lambda_\rho}
\langle \Omega_X 0 \lambda_{b_1} | U_X | J_X m_X \rangle k^{L_X}
\langle \Omega_X 0 \lambda_{b_1} | U_X |
\langle \Omega_{b_1} 0 \lambda_\omega | U_{b_1} | 1 , m_{b_1}=\lambda_{b_1} \rangle q^{L_{b_1}}
\left\{|J_X m_X \rangle + \epsilon_\gamma \epsilon_R P_X (-1)^{J_X-m_X} |J_X\;-m_X \rangle \right\}
\langle \Omega_\omega 0 \lambda_\rho | U_\omega | 1 , m_\omega=\lambda_\omega \rangle u^{L_\omega}
</math>
\langle \Omega_\rho 0 \lambda_\rho | U_\rho | J_\rho , m_\rho=\lambda_\rho \rangle C_\rho(L_\rho) v^{L_\rho}
::<math>
\langle \Omega_{b_1} 0 \lambda_\omega | U_{b_1} | 1 , m_{b_1}=\lambda_{b_1} \rangle
\langle \Omega_\omega 0 \lambda_\rho | U_\omega | 1 , m_\omega=\lambda_\omega \rangle
\langle \Omega_\rho 0 \lambda_\rho | U_\rho | J_\rho , m_\rho=\lambda_\rho \rangle C_\rho(L_\rho)
</math>
::<math>
\langle J_X m_X (\epsilon_\gamma \epsilon_R)|V|
\epsilon_\gamma ; J_R \lambda_R \epsilon_R ; t, s; \Omega_0 \rangle
</math>
</math>