Calibration and time resolution using $\mu$-pair events

Muons from 150 $\times$ $10^3$ $\mu$-pair events corresponding to an accumulated luminosity of 250 $pb^{-1}$ were used for calibration [58]. Electrons in Bhabha events are frequently accompanied by back splash from ECL behind the TOF counters and thus can not be used to simulate hadrons reliably.
The following empirical formula was used for the time walk correction to get a precise observed time,

$$T^{twc}_{obs} = T_{raw} - \left(\frac{z}{V_{eff}} + \frac{S}{\sqrt{Q}} +F(z) \right)$$ (3)

where $T_{raw}$ is the PMT signal time, $z$ is the particle hit position on a TOF counter, $V_{eff}$ is the effective velocity of light in the scintillator, Q is the charge of the signal, S is the coefficient of time walk, and

$$F(z) = \sum_{n=0}^{n=5} A_n z^n.$$ (4)

The coefficients, $1/V_{eff}, S$ and $A_n$ for n = 0 to 5, were determined by minimizing residuals defined as $\delta t = T_{obs}^{twc} - T_{pred}$ for all available TOF hits. Here, $T_{pred}$ is the time of flight predicted using the track length calculated from the fit to hits in CDC. The optimization can be done for all PMTs together or for each PMT separately.
Figure shows time resolutions for forward and backward PMTs and for the weighted average time as a function of $z$. The resolution for the weighted average time is about 100 ps with a small $z$ dependence. This satisfies the design goal, and we expect further improvement. The distribution of fitted residuals $\delta t$ shows an oscillatory behavior of amplitude of $\pm$25 ps as a function of $z$ after calibration. This indicates that the 5-th order polynomial $F(z)$ does not match the data well, and a better choice of the formula may give us further improvement.

Figure: Time resolution for $\mu$-pair events.