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Amplitudes for the Exotic b1π Decay (view source)
Revision as of 04:35, 12 August 2011
, 04:35, 12 August 2011no edit summary
== Application ==
== Application ==
=== Production ===
==== Proton-Reggeon vertex ====
The amplitude of target proton's emission of an exchange particle, a reggeon, in particular direction and helicity projections can be written as:
<table>
<tr>
<td><math>
\langle \Omega_R \lambda_R \lambda_p | W | J_T m_T \rangle
=
\langle \Omega_R \lambda_R \lambda_p
| J m \lambda_R \lambda_p \rangle \langle J m \lambda_R \lambda_p |
W | J_T m_T \rangle
</math></td>
<td>
transition amplitude for <math>p \rightarrow R + p'</math>
in the direction <math>\Omega_R</math> w.r.t. the coordinate
system defined in the resonance RF.
</td>
</tr>
<tr>
<td><math>
=\sqrt{\frac{2J+1}{4\pi}} D_{m_T (\lambda_R-\lambda_p)}^{J_T *} (\Omega_R,0) w_{\lambda_R , \lambda_p}^{J_T}
</math></td>
<td>
follows from relations given above
</td>
</tr>
</table>
==== Photon-Reggeon-Resonance vertex ====
Consider the production of the resonance from the photon and reggeon in the reflectivity basis, the eigenstates of the reflectivity operator. (This operator is a combination of parity and <math>\pi</math> rotation about the normal to the production plane (usually y axis.)
<br><math>\mathbb{R}| J m \rangle = P(-1)^{J-m} | J \; -m \rangle </math>
The eigenstates of the reflectivity operator are formed as follows:
<br><math>| J m \epsilon \rangle = | J m \rangle + \epsilon P (-1)^{J-m} | J \; -m \rangle </math>
<br> such that
<br><math>\mathbb{R}| J m \epsilon \rangle = \epsilon | J m \epsilon \rangle </math>
The photon linear polarization states turn out to be eigenstates of reflectivity as well:
<br>Let x (y) polarization states be denoted with - (+)
<br><math>|\mp\rangle = \sqrt{\frac{\pm 1}{2}} \left( |1 -1\rangle \mp |1 +1\rangle \right)</math>
<br><math>\mathbb{R}|\mp\rangle = \mp 1 |\mp\rangle </math>
Since the production Hamiltonian should commute with reflectivity:
<br><math>V=\mathbb{R}^{-1} V \mathbb{R}</math>
<br><math>
\langle J m \epsilon|\mathbb{R}^{-1} V \mathbb{R}|
\mp ; J_R \lambda_R \epsilon_R ; t, s; \Omega_0 \rangle =
\epsilon (\mp 1) \epsilon_R \langle J m \epsilon|V|
\mp ; J_R \lambda_R \epsilon_R ; t, s; \Omega_0 \rangle
</math>
Acting with the reflectivity operator on initial and final state brings out the reflectivity eigenvalues of the
resonance, photon and reggeon. This result leads to a constraint:
<br><math>\epsilon = \mp \epsilon_R</math>
=== Decay ===
<math>
<math>
<math>
<math>
A^{J_X}=\sum_{\lambda_{b_1},\lambda_\omega,\lambda_\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_X 0 \lambda_{b_1} | U_X | 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_{b_1} 0 \lambda_\omega | U_{b_1} | 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_\omega 0 \lambda_\rho | U_\omega | 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}
\langle \Omega_\rho 0 \lambda_\rho | U_\rho | J_\rho , m_\rho=\lambda_\rho \rangle C_\rho(L_\rho) v^{L_\rho}
</math>
</math>