Event Kinematics Reconstruction

The ZEUS coordinate system is shown in figure 2.6.

Figure 2.6: Schematic view of the ZEUS coordinate system. The orientation of the $Z$ axis is given by the proton beam direction while the $X$ axis points towards the HERA machine center.

The reconstruction of the kinematic variables for neutral current scattering can easily be done if the outgoing positron scattering angle $\theta$ and energy $E$ are measured. Equations (2.1), (2.4) and (2.5) then read

$$Q^2 = 2 A E (1 + \cos \theta)$$ (18)

$$y = 1 - \frac{E}{2 A} ( 1 - \cos \theta )$$ (19)

$$x = \frac{A}{P} \frac{E ( 1 + \cos \theta )}{(2 A - E (1 - \cos \theta))}$$ (20)

with $P$ the energy of the incoming proton and $A$ the energy of the incoming positron. From equation equations (2.18) and (2.19) one can easily derive:

$$Q^2 = \frac{p_{Te}^2}{1-y}$$ (21)

where $p_{Te}$ is the transverse momentum of the scattered positron. Note that the problem is over determined, there are four independent measurement variables, the energy and direction of the scattered lepton and the hadronic system, but only two variables are needed to fully describe the kinematics of the interaction. It is therefore possible to determine the kinematics from any two of the measurement variables [13].

For charged current positron proton scattering the neutrino escapes undetected, so the only detectable particles come from the hadronic final state. In general the struck quark is ejected from the proton and will together with the remnant of the proton hadronize into a final state which typically will consist of a jet proximately in the direction of the struck quark and a jet in the proton remnant direction.

In this case we can determine the kinematic variables from the energy and momenta of the final state particles. Denoting the sum of the four momenta of the final state particles by $p^\prime$ then the momentum transfer vector is given by $q = (p - p^\prime)$. The variable $y$ then follows from

$$y = \frac{p\cdot(p-p^\prime)}{p\cdot k}$$

$$= \frac{P \sum_h(E_h-p_{Zh})}{2PA}$$

$$= \frac{\sum_h(E_h-p_{Zh})}{2A}$$ (22)

where the sum runs over all final state particles in the hadronic system. Because of conservation of transverse momentum we can use equation (2.21) to determine ${Q^2}$ by replacing ${p_{Te}^2}$ by ${p_{Th}^2}$:

$$p_{Th}^2 = \left(\sum_h p_{Xh}\right)^2 + \left(\sum_h p_{Yh}\right)^2$$ (23)

where again the sum runs over all final state particles in the hadronic system. Finally the value of ${x}$ can be obtained from equation (2.6). In this way we arrive at the Jacquet-Blondel variables [14] for the reconstruction of the kinematic variables:

$${y_{\mbox{\em jb}}} = \frac{\sum_i ( E_i - {P_z}_i )}{ 2 * {E_e}}$$ (24)

$${Q^2_{\mbox{\em jb}}} = \frac{(\sum_i {P_x}_i)^2 + (\sum_i {P_y}_i)^2}{ 1 - {y_{\mbox{\em jb}}}}$$ (25)

$${x_{\mbox{\em jb}}} = \frac{{Q^2_{\mbox{\em jb}}}}{ 4 * {E_e}* {E_p}* {y_{\mbox{\em jb}}}}$$ (26)

For an ideal detector and if all particles of the hadronic system were measured the above formulas would exactly return the event kinematics variables. In reality however the energy of the particles can only be determined with a finite precision. Moreover the particles lose energy while traversing material before hitting the calorimeter. For this reason the Jacquet-Blondel estimators tend to return a smaller value than the true one for ${Q^2}$ and ${x}$.

It is interesting to note that the Jacquet-Blondel estimators are not so sensitive to particles escaping through the beam pipe hole, as these particles usually carry small ${P_t}$ and $({{E_{\mbox{\em tot}}}\, - \, {P_z}})$.

It is useful to introduce another variable which is the hadronic angle . In the quark parton model gives the polar angle of the struck quark:

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Figure 3.4: Schematic view of the front face of the RCAL.

The entire UCAL thus consists of cells. The scintillation light of each cell is guided to the back of the calorimeter via two wavelength shifter ( WLS) guides, one on either side of each cell. Each WLS is read out by a photomultiplier ( PMT). To reduce the effects of the cracks between modules, in particular the generation of Cherenkov light in the WLS, sheets of of lead are placed between all the modules.

The energy resolution of the UCAL is for electromagnetic showers and for hadronic showers where the shower energy $E$ is in .

The timing resolution of each individual channel is given by

where is the energy recorded in the channel (see [18]).

The radioactivity of the depleted Uranium causes a constant current drawn by the PMTs. The current is proportional to the cell size and used to calibrate the absolute energy scale of the calorimeter [19].

The UCAL readout system consists of two systems, one for the FLT and one for the SLT and higher level triggers. The signal for the FLT is split off on the detector and analogue sums of several cells are made. The SLT readout system has the higher precision of the full digitized readout.