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Amplitudes for the Exotic b1π Decay (view source)
Revision as of 20:26, 13 August 2011
, 20:26, 13 August 2011no edit summary
Consider the t-channel production of a resonance from the photon and reggeon in the reflectivity basis, consisting of plane-wave states constructed to be eigenstates of the reflectivity operator. This turns out in the case of the photon to correspond to the usual linear polarization basis |x> and |y>. Let the x (y) linear polarization states be denoted as ε=- (ε=+).
Consider the t-channel production of a resonance from the photon and reggeon in the reflectivity basis, consisting of plane-wave states constructed to be eigenstates of the reflectivity operator. This turns out in the case of the photon to correspond to the usual linear polarization basis |x> and |y>. Let the x (y) linear polarization states be denoted as ε=- (ε=+).
:<math>|\varepsilon\rangle = \sqrt{\frac{-\varepsilon}{2}} \left( \left|1\; -1\right\rangle +\varepsilon \left|1\; +1\right\rangle \right)</math>
:<math>|\epsilon\rangle = \sqrt{\frac{-\epsilon}{2}} \left( \left|1\; -1\right\rangle +\epsilon \left|1\; +1\right\rangle \right)</math>
:<math>\mathbb{R}|\varepsilon\rangle = \varepsilon |\varepsilon\rangle </math>
:<math>\mathbb{R}|\epsilon\rangle = \epsilon |\epsilon\rangle </math>
The strong interaction Hamiltonian respects reflectivity, so the production operator ''V'' should commute with ''R''.
The strong interaction Hamiltonian respects reflectivity, so the production operator ''V'' should commute with ''R''.
</math>
</math>
where <math>\alpha=-\phi</math> - the rotation angle to return from production plane to lab xz plane.
where <math>\alpha=-\phi</math>: the rotation angle to return from production plane to lab xz plane. Thus, the production amplitude becomes:
:<math>
\frac{1}{\sqrt{2}}
\langle J_X m_X (\epsilon_\gamma \epsilon_R)|V|
\sqrt{-\epsilon} \left( e^{-i\alpha}\left|1\; -1\right\rangle +
\epsilon e^{i\alpha}\left|1\; +1\right\rangle \right) ;
J_R \lambda_R \epsilon_R ; t, s; \Omega_0 \rangle
</math>
== Decay of t-channel resonance X==
== Decay of t-channel resonance X==
</math>
</math>
== Assembly of the full amplitude ==
:<math>
:<math>
</math>
</math>
::<math>
::<math>
\frac{1}{\sqrt{2}}
\langle J_X m_X (\epsilon_\gamma \epsilon_R)|V|
\langle J_X m_X (\epsilon_\gamma \epsilon_R)|V|
\epsilon_\gamma ; J_R \lambda_R \epsilon_R ; t, s; \Omega_0 \rangle
\sqrt{-\epsilon_\gamma} \left( e^{-i\alpha}\left|1\; -1\right\rangle +
\epsilon_\gamma e^{i\alpha}\left|1\; +1\right\rangle \right) ;
J_R \lambda_R \epsilon_R ; t, s; \Omega_0 \rangle
</math>
The intensity would then require an incoherent summation of amplitudes with laboratory x and y polarization.
:<math>
\frac{1+f}{2}\left|\sum_{L_X} A_{L_X\;-1\;\epsilon_R}^{J_X}\right|^2 +
\frac{1-f}{2}\left|\sum_{L_X} A_{L_X\;+1\;\epsilon_R}^{J_X}\right|^2
</math>
</math>
where f is the polarization fraction varying from 1, 100% x-polarized, to 0, unpolarized.