The Mu3e tracker faces the challenge of high particle rates and the need to minimize the material budget along the particles trajectories. Novel silicon High-Voltage Monolithic Active Pixel Sensor (HV-MAPS) which can be thinned down provide time resolutions in the order of nanoseconds and good spacial resolution. Hence, they are an ideal choice for the experiment’s tracker.
2.4.1 The Sensors: HV-MAPS
InHV-MAPS, transistors are integrated into the diode which provides almost full fill factor. This low voltage part is shielded by deep N-wells from the charge-collecting diodes. Order of 60 V reverse biasing of the diodes results in thin, about 10 µm thick, depleted regions with high electric drift fields. This leads to fast charge collection ofO(1 ns) and allows to thin down the substrate to 50 µm.
2.4. TRACKING
The MuPix Sensor
A dedicatedHV-MAPSis being developed for the Mu3e experiment [34]. This MuPix sensor is fabricated currently in the commercial AMS H18 180 nm High-Voltage CMOS process. Its design goal has an active area of roughly 2 cm×2 cm with pixel size of about 80 µm×80 µm.
At least the first amplification stage is located inside the pixel, potentially also the comparator stage. Each pixel is connected to the periphery which hosts the signal processing and digital state machine. A 8 ns timestamp is assigned to each pixel hit. Up to four 1.25 Gbps Low Voltage Differential Signalling (LVDS) links per chip transmit the zero-suppressed pixel data. A low number of signal and slow control lines is required to reduce the material budget of sensor’s support structure. A power consumption below 300 mW/cm2is targeted.
MuPix8, the sensor’s most recent R&D version, has an active size of 2 cm×1 cm and hosts different matrices for the signal propagation from pixel to the periphery. High efficiencies above 99 % at negligible noise rates below 1 Hz/pixel have been demonstrated [69]. This sensor shows a time resolution ofO(14 ns) which is consistent to its predecessors [69, 70].
2.4.2 The Tracker Support Structure
The pixel sensors are tab-bonded onto Kapton-aluminium flex print High Density Intercon-nect (HDI) which provides power together with signal and slow control lines. Multiple sensors along the beam direction are combined to a ladder on the same HDI. The ladders are reinforced with 25 µm thick polyimide foil. In the outer two tracking layers additional v-folds are added to enhance the mechanical stability. At the same time, they are used as channels for addi-tional flow of gaseous Helium which is used to cool the sensors. The radiation length of the combination of the sensors with the support structure of one layer is projected to be about 0.1 %X0.
The pixel ladders are combined to modules which are mounted between support rings on both sides.
2.4.3 Triplet-Based Track Finding
The track finding in the experiment is based on triplets of pixel hits which are combined to actual tracks in a second step.
Triplets
Due to the small pixel size and low particle momenta, the tracking uncertainties are dominated by Multiple Scattering (MS) (see Equation 2.7). Furthermore almost all material, which causes scattering, is located around the tracking planes. Motivated by these two observations, a helix fit which treatsMSin the pixel detector planes as the only uncertainty has been developed by the Mu3e collaboration [67, 71]. Variations of the same algorithm are employed for online reconstruction required for event filtering and offline data analysis. The algorithm is based on triplets of hits, which can be fit in parallel. Thus the method is suited for online track finding.
CHAPTER 2. THE MU3E EXPERIMENT
Figure 2.6:Two helices through a hit triplet (1,2,3) with scattering anglesΦMSandΘMSin the ma-terial in the plane of the central (2) hit. The magnetic field is parallel to the z-axis which is aligned with the beam direction. The left side shows a projection in a transverse plane, the right side a pro-jection on the longitudinal (z-s) plane, where s is the path length along the trajectories. Adapted from [67].
The reconstructed tracks consist of combinations of multiple triplets of hits in successive detector layers. Figure 2.6 illustrates two helices through the triplet hits with scattering in the material in the plane of the central hit (2). In first order, multiple scattering in the material of thin tracking layers only alters a particle momentum’s direction but not the amplitude, thus the total energy. Hence the three-dimensional bending radius in the homogeneous magnetic field
R23D =rt,122 + z122
Φ212 =r2t,23+ z232
Φ223 (2.8)
is conserved. The scattering is expressed by the scattering anglesΦMSandΘMSin the trans-versal and longitudinal plane respectively. According toMStheory (see subsection 2.2.1), the scattering angles distribute around zero with variances ofσθ2 = σMS2 andσ2ϕ = σMS2 /sin2θ.
The algorithm minimizes
χ2(R3D) = ΦMS(R3D)2
σ2ϕ +ΘMS(R3D)2
σ2θ (2.9)
by using a linearization around an approximate solution withrt,12 = rt,23. More than one solution exists for different numbers of half turns of the helix.
Track Finding
The triplets are built from pre-selected hits in a reconstruction time frame. Currently, recon-struction frames of 50 ns are used, which is roughly aligned to the silicon pixel’s time resolu-tion ofO(14 ns). Track candidates are built by combining valid multiple triplets which yields a common translates directly into a particle’s momentum, is the weighted average of all triplet radiiRi.