The (hopefully) Definitive Guide to Coordinate Systems and Transformation in the Low Energy Hall A Experiments

In the upcoming polarized beam Hall A experiment the polarization of a detected proton (or deuteron) is measured in the Focal Plane Polarimeter (FPP) installed in the detector stack of the left HRS. As the name implies the FPP measures the polarization components of the detected particle in the focal plane using the coordinate system of the FPP (see 2.4). The physics observables, however are calculated by theorists in different coordinate system depending on the reaction under study. It is therefore necessary to perform a coordinate transformation on the extracted polarization in order to allow comparison with the theory (it should also be noted that since the definition of the FPP coordinate system varies per event, the calculation needs to be performed on an event by event basis, or at least for small bins, before the extraction of the polarization components). The transformation from the frame in which the physics observables are reported (henceforth the ”physics frame”) and the FPP proceeds through the transport (HRS) frame and the spin transport matrix which rotates the spin of the detected particle on the magnetic field of the HRS. In total the polarization of the detected particle in the FPP may be written as: