Changes

== General Relations ==

= General Relations =

=== Angular Distribution of Two-Body Decay ===

== 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>.   

where <math>\lambda=\lambda_1-\lambda_2</math>,and the double-stacked symbols are Clebsh-Gordon coefficients.  The product of CG coefficients in the second brackets on the right-hand side represent the overlap between the basis vectors in the helicity and ''L,S'' basis, which turns out to be independent of ''m'', as required by rotational invariance.  This overlap integral is somewhat lengthy to calculate, but the result turns out to be fairly simple, as shown above.  This expression holds regardless of what axis is used to define the quantization direction for m, but whatever choice is made must serve as the z-axis of the reference frame in which the plane wave direction <math>\Omega</math> is defined.

where <math>\lambda=\lambda_1-\lambda_2</math>,and the double-stacked symbols are Clebsh-Gordon coefficients.  The product of CG coefficients in the second brackets on the right-hand side represent the overlap between the basis vectors in the helicity and ''L,S'' basis, which turns out to be independent of ''m'', as required by rotational invariance.  This overlap integral is somewhat lengthy to calculate, but the result turns out to be fairly simple, as shown above.  This expression holds regardless of what axis is used to define the quantization direction for m, but whatever choice is made must serve as the z-axis of the reference frame in which the plane wave direction <math>\Omega</math> is defined.

=== Isospin Projections ===

== Isospin Projections ==

One must also take into account the various ways that the isospin of the daughters can add up to the isospin quantum numbers of the parent, requiring a term:

One must also take into account the various ways that the isospin of the daughters can add up to the isospin quantum numbers of the parent, requiring a term:

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 ===

== 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.

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.

:<math>\mathbb{R}| J m \epsilon \rangle =  \epsilon (-1)^{2J} | J m \epsilon \rangle </math>

:<math>\mathbb{R}| J m \epsilon \rangle =  \epsilon (-1)^{2J} | J m \epsilon \rangle </math>

== Application ==

= Applications =

=== Production ===

== Production ==

==== 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.  

Consider the production of the resonance from the photon and reggeon in the reflectivity basis, the eigenstates of the reflectivity operator.