Comment on Resonances

["Curtis A. Meyer" <cmeyer@ernest.phys.cmu.edu>]

Dear Offline Colleagues -

   durning the offline meeting yesterday (June 5, 2007), there
was a comment about being able to know the history of particles
coming from broad resonaces such as the b1 and rho in the 
Monte Carlo. During the meeting, I commented that in doing
Partial Wave Analysis, one would not put these broad resonances
into the Monte Carlo sample, rather one would throw the appropriate
n-body phase space and then let the fit decide what the relevant physics
was (including resonances).

   There is one situation where one might want to put the resonances
in. That would be when we are testing fits on Monte Carlo Data. In that
case, we would need to weight the "data set" by the appropriate physics
weights. However, these weights WOULD INCLUDE all possible decay
chains through the data---that is what the physics would say. As an example,
consider the production of a neutral eta_2 (spin2 isospin zero) state X
which then decays to an a2 and a pi

     gamma p -> X(0) p
                          X(0) -> a2 pi_1
                                  a2 -> rho pi_2
                                            rho -> pi_3 pi_4

      So the final state is four pions. Let us for simplicity assume that we
have pi+ pi- pi0 pi0. There are actually four ways that the event could
have been produced.

                         [a2-zero   pi-zero]      rho+ pi-     
                                                            rho- pi+
                         [a2-plus   pi-minus]    rho+ pi0
                         [a2-minus pi-plus]      rho-  pi0

The physics weight for this event would include all four possible decay chains
with appropriate isospin factor and resonance weights (e.g. Breit Wigner). It
doe not make sense to say that a particular pion came from the a2 or from the 
rho, there is only a probability that one of them did.  These also interfere with 
each other, so even the probability is not overly meaningful. There is an amplitude
that includes all of these and that has to be right.


If we go to the situation of narrow resonances---the pi0, eta, omega, etaprime and phi,
then we can in some sense treat these as final state particles and to first order ignore
the combinatorics. However, this may not be true for the omega in all situations. In Crystal
Barrel when we looked at 

   p-bar p -> omega pi0 pi0  and omega -> pi0 gamma

we actually had to write the amplitudes with all three pi0s having the
possibility of coming from the omega. There were regions where this
was ambigious.

   Finally, the fact that genr8 allows one to impose a particular resonance path
does not meant that is true physics.  So, I come back to my statement from
yesterday in that narrow resonances (as defined above) can have a history.
In general, nothing else does.

   -- Curtis
-- 
Professor Curtis A. Meyer        Department of Physics
Phone:  (412) 268-2745          Carnegie Mellon University
Fax:    (412) 681-0648            Pittsburgh PA 15213-3890
cmeyer@ernest.phys.cmu.edu  http://www.curtismeyer.com/