Parity Relations in the Reflectivity Basis
Paul Eugenio
and
Curtis A. Meyer
September 21, 2000
We want to consider the photo-production process
and to derive partity relations between the production strengths, ,
as seen in the reflectivity basis. These will then be combined with
the coresponding decay amplitudes as cmputed in the reflectivity
basis to allow us to compute event weigths for our PWA
analysis. We start initially with conservation of parity in the
helicity frame. For the process
, we know
that the following relation ship holds.
The refer to the helicities of the indicated particles, the
refer to the spins of the particles, and the refer to the
parity of the particles. This can be simplified by noting first that three
of the parities can be eliminated:
Next, we can eliminate a lot of the spin factors as:
and then we know that the naturality of is given as:
Finally, we can also simplify the term involving . For the
case of photo-production we know that
and also
that
. This means that
,
and therefore
. We can now simplify the
parity relationship to:
For the case of spin-flip ,
, while for
the case of spin-non-flip ,
which yields:
We can now write these production strengths for the
states in the reflectivity basis for a produced particle of naturality
in a reflectivity state . In this case,
Let us now take the case where we only have a spin-flip contributing. In addition,
helicity conservation will limit the values of to be
.
In this case
, and we can write that:
For the case of only spin-non-flip contributions, we have that
, and we can write that:
The last part is to rotate the spin-density matrix of the photon from
the helicity frame,
to the
reflectivity basis
. Where
. To do this, we recall that the basis states can be
transformed as:
There is also a simple relation between the reflectivity states and the
linear polarization states,
and
. We can now multiply out to obtain
the four elements of the spin-density matrix in the reflectivity basis.
We also know that
and that
. Using this, we can simplify the spin-density
matrix as follows:
For the case of an unpolarized photon, we know that
and that
.
This means that the spin-density matrix is as expected:
For the case of linearly polarized photons, we know that
where is the angle between the
polarization vector and the normal to the production plane as seen in the
Gottfried Jackson frame. In addition,
and
. This allows us to simplify the
spin-density matrix to:
Now, for the case of production via exchange, there are
four complex production amplitudes,
. There are two possible
spin-flips for each of the two reflectivity. However, parity will reduce this down to
two complex production strengths,
and
. If we write
and
as the decay
amplitudes for the two reflectivity states of the , then the weight for a
particular event is given as:
Using the density matrix from above and noting the fact that
, we can somewhat simplify the
above expression to be proportional to:
Where is the product of times .
Curtis A. Meyer
2000-09-21