Amplitudes for the Exotic b1π Decay

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Angular Distribution of Two-Body Decay
Let's begin with a general amplitude for the two-body decay of a state with angular momentum quantum numbers J,m. Specifically, we want to know the amplitude of this state having daughter 1 with trajectory $$\Omega=(\phi,\theta)$$. We can also describe the angular momentum between the daughters as being L and spin sum as s. Alternatively, we will label the daughters as having helicities of $$\lambda_1$$ and $$\lambda_2$$ or direction of decay (specified by daughter 1) of

Isospin Projections
One must also take into account the various ways isospin of daughters can add up to the isospin quantum numbers of the parent, requiring a term:

$$ C^{a,b} = \left(\begin{array}{cc|c} I^a   &  I^b   &  I \\ I_z^a & I_z^b & I_z^a+I_z^b \end{array}\right) $$

where a=1 and b=2, referring to the daughter number. Because an even-symmetric angular wave function (i.e. L=0,2...) imply that 180 degree rotation is equivalent to reversal of daughter identities, a,b becoming b,a on must write down the symmetrized expression:

$$ C(L)=\frac{1}{\sqrt{2}} \left[ C^{a,b} + (-1)^L C^{b,a} \right] $$

Application
$$ \langle \Omega_X 0 \lambda_{b_1} | U_X | J_X m_X \rangle =\sum_{L_X} \left[ \sqrt{\frac{2J_X+1}{4\pi}} D_{m_X \lambda_{b_1}}^{J_X *}(\Omega_X,0) \right] \left[ \sqrt{\frac{2L_X+1}{2J_X+1}} \left(\begin{array}{cc|c} L_X & 1            & J_X \\ 0   & \lambda_{b_1} & \lambda_{b_1} \end{array}\right) \right] a_{L_X}^{J_X} $$

$$ \langle \Omega_{b_1} 0 \lambda_\omega | U_{b_1} | 1, m_{b_1}=\lambda_{b_1} \rangle =\sum_{L_{b_1}} \left[ \sqrt{\frac{2J_{b_1}+1}{4\pi}} D_{m_{b_1}=\lambda_{b_1} \lambda_\omega}^{1 *}(\Omega_{b_1},0) \right] \left[ \sqrt{\frac{2L_{b_1}+1}{2J_{b_1}+1}} \left(\begin{array}{cc|c} L_{b_1} & 1             & 1 \\ 0       & \lambda_\omega & \lambda_\omega \end{array}\right) \right] b_{L_{b_1}} $$

$$ \langle \Omega_\omega 0 \lambda_\rho | U_\omega | 1, m_\omega=\lambda_\omega \rangle =\sum_{L_\omega J_\rho} \left[ \sqrt{\frac{2J_\omega+1}{4\pi}} D_{m_\omega=\lambda_\omega \lambda_\rho}^{1 *}(\Omega_\omega,0) \right] \left[ \sqrt{\frac{2L_\omega+1}{2J_\omega+1}} \left(\begin{array}{cc|c} L_\omega & 1           & 1 \\ 0        & \lambda_\rho & \lambda_\rho \end{array}\right) \right] c_{L_\omega J_\rho} $$

$$ \langle \Omega_\rho 0 \lambda_\rho | U_\rho | J_\rho, m_\rho=\lambda_\rho \rangle =\sum_{L_\rho} \left[ \sqrt{\frac{2J_\rho+1}{4\pi}} D_{m_\rho 0}^{J_\rho *}(\Omega_\rho,0) \right] \left[ \sqrt{\frac{2L_\rho+1}{2J_\rho+1}} \left(\begin{array}{cc|c} L_\rho & 0 & J_\rho \\ 0     & 0 & 0 \end{array}\right) \right] d_{L_\rho} =\sum_{L_\rho} \sqrt{\frac{2L_\rho+1}{4\pi}} Y_{m_\rho}^{J_\rho *}(\Omega_\rho) d_{L_\rho} $$

$$ A^{J_X}=\sum_{\lambda_{b_1},\lambda_\omega,\lambda_\rho} \langle \Omega_X 0 \lambda_{b_1} | U | J_X m_X \rangle C_X(L_X) k^{L_X} \langle \Omega_{b_1} 0 \lambda_\omega | U | 1, m_{b_1}=\lambda_{b_1} \rangle C_{b_1}(L_{b_1}) q^{L_{b_1}} \langle \Omega_\omega 0 \lambda_\rho | U | 1, m_\omega=\lambda_\omega \rangle C_\omega(L_\omega) u^{L_\omega} \langle \Omega_\rho 0 \lambda_\rho | U | J_\rho, m_\rho=\lambda_\rho \rangle C_\rho(L_\rho) v^{L_\rho} $$

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