Mathematical Framework for Fast and Rigorous Track Fit for the ZEUS Detector

The magnetic field in the central region of the ZEUS detector was produced by a thin superconducting solenoid. The field had a strength of 14.3 kGauss at the center and was directed parallel to the proton beam. The barrel MVD and CTD were located in the field which was almost homogeneous with a small radial component far from the center. Forward trackers were placed outside of the solenoid or close to its edge where the field is inhomogeneous.

We consider a mathematical framework for a rigorous approach to a common track fit, which can be performed with tracks including all inner tracking components or with any combination of them. The approach offers a rigorous treatment of field inhomogeneity, multiple scattering and energy loss. The track fitting procedure makes use of the Kalman filter technique and we discuss how to optimize computations and make the fitting procedure fast.

The ZEUS coordinate system is a right-handed Cartesian system, with the z-axis pointing in the proton beam direction (forward) and the x-axis pointing to the center of the HERA ring. The coordinate origin is at the nominal interaction point.

The barrel (BMVD) and forward (FMVD) section of the MVD includes 600 and 112 sensors, respectively [2]. A sensor is a silicon single-sided strip detector with a readout pitch of 120 µm which includes five innermost strips for capacitive charge division. The ZEUS MVD has 307,200 and 53,730 readout channels in the barrel and forward sections, respectively.

The barrel section, centered at the interaction point, is about 63 cm long. The silicon sensors are arranged in three concentric cylindrical layers with radii about 5 cm, 8 cm and 12 cm. Two back to back sensors in a layer provide measurements of nominal r -φ and z position. The FMVD is composed of four transverse disks of 14 wedges each, which extend the angular coverage down to 7 • from the beam line. Each wedge has two sensor layers separated by approximately 8 mm in z-direction. They are mounted back to back, such that the angle between strips is 2 × 13 • .

The CTD [2] is a cylindrical drift chamber, with a sensitive volume approximately 2m in length and 0.4 (1.6m) in inner (outer) diameter. The CTD wires are arranged into nine concentric superlayers numbered consecutively from the inside out. The odd-numbered superlayers have sense wires running parallel to the chamber axis (i.e. z-axis) while those in the even-numbered superlayers have a 5 • stereo angle. We denote sense wires in corresponding superlayers as axial and stereo, respectively. Each superlayer contains eight sense wire layers -there are 4608 sense wires in total. A set of eight sense wires is surrounded by field wires, azimuthally dividing a superlayer into cells of polygonal shape. Each sense wire is read out by a flash ADC and, finally a drift distance is evaluated for a hit. All axial wires in superlayer one and the odd numbered wires in superlayer three and five (in total 704 wires) are additionally equipped with the z-by-timing system, which measures z position of a hit.

The STT uses straw drift chambers with 7.5 mm diameter capton tubes of varying length from 20 cm to 75 cm. There are in total 10,944 wires in 48 wedge shaped sectors. Each wedge covers an azimuthal span of 60 • . Each sector consists of 3 layers of straws perpendicular to the z-axis. A track crossing the STT nominally delivers 24 drift time measurements.

The likelihood function of a track measurement has a meaning regardless of the details of any fitting method. The maximum-likelihood estimator is efficient in the sense that no other unbiased estimator has smaller variances. A track model which is appropriate for the likelihood function, together with a given method of track fit, may produce an efficient estimate of parameters. A general point of view of the information delivered by a tracker can help to interpret behavior of variances of fitted parameters and hit residuals. We can model a multi-component tracker by a set of track detecting elements and intermediate blocks of passive material, which are located in a static magnetic field. Track parameters in the detector element k are described by a vector x k . For the case of a three-dimensional fit, the dimension of the vector,x k , is 5. The track measurement in the tracker element k, i.e.

…(Full text truncated)…