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Line 77: − + − <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> −
Amplitudes for the Exotic b1π Decay (view source)
Revision as of 20:54, 12 August 2011
, 20:54, 12 August 2011no edit summary
</math>
</math>
It should be kept in mind that this <math>C(L)</math> is only applicable for particle pairs in the same isospin multiplet.
It should be kept in mind that this <math>C(L)</math> is only applicable for particle pairs in the same isospin multiplet.
=== Reflectivity ===
Beside rotational invariance, parity is also a good symmetry of strong hadron dynamics. In the case described above of the decay of a single particle at rest into two daughters, parity conservation places constraints between different final state amplitudes. Instead of considering the parity operator directly, it is convenient to consider the reflectivity operator ''R''. Reflectivity is the product of parity with a 180 degree rotation about the y axis. The advantage of using this more complicated operator to express the constraints of parity is that a general two-particle plane wave basis can be constructed out of eigenstates of reflectivity, whereas a complete plane-wave basis of parity eigenstates is possible only in the restricted case that daughters 1 and 2 are identical. Regardless of the additional rotation, the basic constraint of reflectivity conservation is nothing more than parity conservation plus rotational invariance.
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>
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.
:<math>| J m \epsilon \rangle = | J m \rangle + \epsilon P (-1)^{J-m} | J \; -m \rangle </math>
such that
:<math>\mathbb{R}| J m \epsilon \rangle = \epsilon (-1)^{2J} | J m \epsilon \rangle </math>
== Application ==
== Application ==
==== Photon-Reggeon-Resonance vertex ====
==== 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.)
Consider the production of the resonance from the photon and reggeon in the reflectivity basis, the eigenstates of the reflectivity operator.
The photon linear polarization states turn out to be eigenstates of reflectivity as well:
The photon linear polarization states turn out to be eigenstates of reflectivity as well: