[Halld-physics] correlations in neutral shower parameters

[Richard Jones]

In laying out the dst format for MC studies, I claimed in an earlier message that the error matrix used to store the errors on neutral hit parameters only needs to contain 3 (bcal) or 2 (fcal) off-diagonal elements.  Our argument follows below.  The starting point for this is a single cluster reconstructed in the bcal or fcal.  No vertex hypothesis has yet been applied to this hit, nor has a momentum vector been assigned to it.  It is just a neutral cluster in a calorimeter.

 1. *bcal* - the set of measured hit parameters which are most loosely coupled in terms of the raw measurements are (r,phi,z,E,t).  A transformation can be used to convert into any other preferred set of hit parameters.  Correlations are (r,z) , (r,E) , and (z,E).
      * phi is decoupled from all of the others.
      * r is directly measured through the radial profile, but couples to z and E implicitly.
      * z is independently measured using timing, but is implicitly correlated with r through fluctuations in the energy deposition for oblique showers, and with E through the position of the shower maximum, and explicitly with E through the attenuation correction.
      * E correlates implicitly to r,z through shower depth variations, and explicitly to z through the attenuation correction.
      * the t of the shower maximum correlates with r,z (and through them with E).  However this correlation is trivial, and will be taken out when the time is projected back from the shower maximum to a reference vertex.  Since the speed of light is independent of E, this correlation of t with r,z is trivial and need not be carried along with the shower.
 2. *fcal* - the set of measured hit parameters which are most loosely coupled in terms of the raw measurements are (x,y,E,t).  Note that the depth z of the shower maximum is not measured in the fcal, so it does not belong in the list, being a pure function of E.  Correlations are (x,y) , (x,E) , and (y,E).
     1. x,y are implicitly correlated with each other by the shape of shower deposition in the fcal, and with E through shower depth.
     2. as for the fcal, the t of the shower maximum is trivially correlated with E,x,y.  I argue that this correlation is trivial and need not be carried along with the shower object for the same reasons as given above under the bcal heading.

Thus, for the bcal we must keep 3 off-diagonal error matrix elements per shower.  For the fcal there are also 3 non-trivial off-diagonal elements, but two of them are related to each other in a simple way:

                x * sigma(y,E) = y * sigma(x,E)

So strictly speaking there are only 2 off-diagonal error matrix elements per shower that must be kept in the fcal.

-Richard J.
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