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Line 154: − − ==== 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>
Amplitudes for the Exotic b1π Decay (view source)
Revision as of 13:26, 12 August 2011
, 13:26, 12 August 2011no edit summary
=== Production ===
=== Production ===
==== 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>
such that
<br><math>\mathbb{R}| J m \epsilon \rangle = \epsilon (-1)^{2J} | 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 - (+)
<math>|\mp\rangle = \sqrt{\frac{\pm 1}{2}} \left( |1 -1\rangle \mp |1 +1\rangle \right)</math>
<math>\mathbb{R}|\mp\rangle = \mp 1 |\mp\rangle </math>
Since the production Hamiltonian should commute with reflectivity:
<math>V=\mathbb{R}^{-1} V \mathbb{R}</math>
<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>
==== Proton-Reggeon vertex ====
==== Proton-Reggeon vertex ====
<tr>
<tr>
<td><math>
<td><math>
\langle \Omega_R \lambda_R \lambda_p | W | J_T m_T \rangle
\langle \Omega_R ; J_R \lambda_R \epsilon_R; J_P \lambda_p | W | J_T m_T \rangle
=
=
\langle \Omega_R \lambda_R \lambda_p
\langle \Omega_R ; J_R \lambda_R \; \mp\epsilon; \textstyle{\frac{1}{2}}\;\lambda_p
| J m \lambda_R \lambda_p \rangle \langle J m \lambda_R \lambda_p |
| \textstyle{\frac{1}{2}}\;m_T \lambda_R \lambda_p \rangle \langle \textstyle{\frac{1}{2}}\;m_T \lambda_R \lambda_p |
W | J_T m_T \rangle
W | \textstyle{\frac{1}{2}}\;m_T \rangle
</math></td>
</math></td>
<td>
<td>
<tr>
<tr>
<td><math>
<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}
=\frac{1}{\sqrt{2\pi}} \left[
D_{m_T (\lambda_R-\lambda_p)}^{\frac{1}{2} *} (\Omega_R,0) \; w_{\lambda_R\; \lambda_p}
\mp
\epsilon P_R (-1)^{J_R-\lambda_R}
D_{m_T (-\lambda_R-\lambda_p)}^{\frac{1}{2} *} (\Omega_R,0) \; w_{\lambda_R\; -\lambda_p}
\right]
</math></td>
</math></td>
<td>
<td>
</table>
</table>