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Amplitudes for the Exotic b1π Decay (view source)
Revision as of 03:22, 25 October 2011
, 03:22, 25 October 2011→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 for having daughter 1 with momentum direction <math>\Omega=(\phi,\theta)</math> in the center of mass reference frame, and helicity <math>\lambda_1</math>, while daughter 2 has direction <math>-\Omega=(\phi+\pi,\pi-\theta)</math> and helicity <math>\lambda_2</math>.
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 for having daughter 1 with momentum direction <math>\Omega=(\phi,\theta)</math> in the center of mass reference frame, and helicity <math>\lambda_1</math>, while daughter 2 has direction <math>-\Omega=(\phi+\pi,\pi-\theta)</math> and helicity <math>\lambda_2</math>.
Let U be the decay operator from the initial state into the given 2-body final state. Intermediate between the at-rest initial state of quantum numbers (qn) J,m and the final plane-wave state is a basis of outgoing waves describing the outgoing 2-body state in a basis of good J,m and helicities. Insertion of the complete set of intermediate basis vectors, and summing over all intermediate J,m gives
Let U be the decay operator from the initial state into the given 2-body final state. Intermediate between the at-rest initial state of quantum numbers (qn) J,M and the final plane-wave state is a basis of outgoing waves describing the outgoing 2-body state in a basis of good J,m and helicities. Insertion of the complete set of intermediate basis vectors, and summing over all intermediate J,m gives
:<math>
:<math>
\langle \Omega \lambda_1 \lambda_2 | U | J M \rangle
\langle \Omega \lambda_1 \lambda_2 | U | J M \rangle
:<math>
:<math>
=\sum_{L,S}
=\sum_{L,S}
\left[ \sqrt{\frac{2J+1}{4\pi}} D_{m \lambda}^{J *}(\Omega) \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}
Acting on a state of good ''J,m'', the reflectivity operator has a particularly simple effect.
Acting on a state of good ''J,m'', the reflectivity operator has a particularly simple effect.
:<math>\mathbb{R}| J M \rangle = P(-1)^{J-m} | J \; -M \rangle </math>
:<math>\mathbb{R}| J M \rangle = P(-1)^{J-M} | J \; -M \rangle </math>
where P is the intrinsic parity of the system.
where P is the intrinsic parity of the system.
The eigenstates of the reflectivity operator are formed out of states of good ''J,M'' as follows.
The eigenstates of the reflectivity operator are formed out of states of good ''J,M'' as follows.
:<math>
:<math>
T_{(f)(i)} =
T_{(f)(i)} =
T_{(\mathbf{q}_\pi \mathbf{q}_\rho \mathbf{q}_\omega \mathbf{q}_{b1} \mathbf{p}_f \epsilon_f)
T_{(\mathbf{q}_\pi \mathbf{q}_\rho \mathbf{q}_\omega \mathbf{q}_{b_1} \mathbf{p}_f \epsilon_f)
(\mathbf{k}_\gamma \epsilon_\gamma \mathbf{p}_i \epsilon_i)}=
(\mathbf{k}_\gamma \epsilon_\gamma \mathbf{p}_i \epsilon_i)}=
</math>
</math>
:::<math>
:::<math>
=\sum_{R,\lambda_R,\epsilon_R;\lambda_{b_1},\lambda_\omega,\lambda_\rho}
=\sum_{R,\lambda_R,\epsilon_R;\lambda_{b_1},\lambda_\omega,\lambda_\rho}
\langle \mathbf{q}_\pi 0 0; \mathbf{q}_\rho \lambda_\rho 0; \mathbf{q}_\omega \lambda_\omega 0; \mathbf{q}_{b1}\lambda_{b1} 0
\langle \mathbf{q}_\pi 0 0; \mathbf{q}_\rho \lambda_\rho 0; \mathbf{q}_\omega \lambda_\omega 0; \mathbf{q}_{b_1}\lambda_{b_1} 0
| UV | \epsilon_\gamma; \lambda_R \epsilon_R; \Omega_0\rangle
| UV | \epsilon_\gamma; \lambda_R \epsilon_R; \Omega_0\rangle
</math>
</math>
q(m) = \sqrt{\left(\frac{m^2+m_1^2-m_2^2}{2m}\right)^2-m_1^2}
q(m) = \sqrt{\left(\frac{m^2+m_1^2-m_2^2}{2m}\right)^2-m_1^2}
</math>
</math>
The functions <math>F_L(q)</math> are the angular momentum barrier factors that are given in the literature.
The functions <math>F_L(q)</math> are the angular momentum barrier factors that are given in the literature. The first few are listed below with <math>z=[q/(197\mathrm{MeV/c})]^2</math>
:<math>\displaystyle F_0(q)=1</math>
:<math>F_1(q)=
\sqrt{\frac{2z}{z+1}}
</math>
:<math>F_2(q)=
\sqrt{\frac{13z^2}{(z-3)^2+9z}}
</math>
:<math>F_3(q)=
\sqrt{\frac{277z^3}{z(z-15)^2+9(2z-5)^2}}
</math>
===Describing s and t dependence===
===Describing s and t dependence===
The T Matrix, written in the photon reflectivity basis can be expanded in the photon's lab frame helicity basis. Temporarily omitting indices not pertaining to the photon:
The T Matrix, written in the photon reflectivity basis can be expanded in the photon's lab frame helicity basis. Temporarily omitting indices not pertaining to the photon:
:<math>
:<math>
T_{\epsilon_\gamma} = \frac{\sqrt{-\epsilon_\gamma}}{2}
T_{\epsilon_\gamma} = \sqrt{\frac{-\epsilon_\gamma}{2}}
\left( T_{-1}e^{-i\alpha} - \epsilon_\gamma T_{+1}e^{i\alpha} \right)
\left( T_{-1}e^{-i\alpha} - \epsilon_\gamma T_{+1}e^{i\alpha} \right)
</math>
</math>