BCal Beam Test Plots, May 8, 2007

Energy Weighting

The timing resolution of the BCal should improve when one begins to group the individual cells into a cluster of cells. Ideally one would like to weight each cell by its own resolution (i.e. multiply by the inverse squared) such that cells with a poorer resolution contribute less to the overall resolution. However, one also needs to know the resolution for each cell to a good degree otherwise the cells which truly have a lower resolution might end up contributing more than they should giving one a worse resolution overall as a result. The next best thing would be to weight each cell by its deposited energy which is directly measured. The ADC signal is proportional to the energy deposit by some calibration constant. The inverse square of the resolution of the cell is also proportional to the energy deposited. Hence, we define the time at each calorimeter end to be

T_{N}={\frac {\sum _{i}t_{{N,i}}E_{{N,i}}}{\sum _{i}E_{{N,i}}}}

T_{S}={\frac {\sum _{i}t_{{S,i}}E_{{S,i}}}{\sum _{i}E_{{S,i}}}}

and the average of T_N and T_S defines the mean time. Each end of the cell should have identical responses and we can assume E_N = E_S. For the case where the beam is at dead center of the module the ADC signals should also be similar except for some gain correction factors. We can then define the mean time and time difference as

T={\frac {1}{2}}{\frac {\sum _{i}(t_{{N,i}}+t_{{S,i}})E_{{i}}}{\sum _{i}E_{{i}}}}

\Delta T={\frac {1}{2}}{\frac {\sum _{i}(t_{{S,i}}-t_{{S,i}})E_{{i}}}{\sum _{i}E_{{i}}}}

respectively. We estimate the contribution to the resolution from jitter in the Master OR signal from the Tagger to be σ_tagger = 113 ps which should be subtracted in quadrature from the constant term of the fit of the mean time resolution to obtain the correct mean time resolution. ΔT and T are the difference and sum of T_N and T_S and therefore the distributions have the same rms. spread. One therefore expects σ_T = σ_ΔT.