Simulation of the Dynamic Inefficiency of the CMS Pixel Detector

The Pixel Detector is the innermost part of the CMS Tracker. It therefore has to prevail in the harshest environment in terms of particle fluence and radiation. There are several mechanisms that may decrease the efficiency of the detector. These are mainly caused by data acquisition (DAQ) problems and/or Single Event Upsets (SEU). Any remaining efficiency loss is referred to as the dynamic inefficiency. It is caused by various mechanisms inside the Readout Chip (ROC) and depends strongly on the data occupancy. In the 2012 data, at high values of instantaneous luminosity the inefficiency reached 2\% (in the region closest to the interaction point) which is not negligible. In the 2015 run higher instantaneous luminosity is expected, which will result in lower efficiencies; therefore this effect needs to be understood and simulated. A data-driven method has been developed to simulate dynamic inefficiency, which has been shown to successfully simulate the effects.


Introduction
The Compact Muon Solenoid (CMS) is one of the two general-purpose detectors that measure the products of high energy particle interactions at the Large Hadron Collider (LHC). The CMS Pixel Detector is a silicon semiconductor detector at the centre of the CMS tracking system. Along with the surrounding Silicon Strip Tracker, it provides precision measurements of the trajectories of charged particles. The Pixel Detector consist of three cylindrical layers called the barrel and two endcap disks at each end, called the forward part of the detector. It lies very close to the interaction point; the mean radius of layers 1, 2 and 3 are 4.4, 7.3 and 10.2 cm, respectively. The barrel is divided into ladders (along the r − φ plane) and rings (along the z axis) as can be seen in figure 1. The intersection of a ladder and a ring is called a module, the basic building block of the detector. A module is made up of 8 or 16 Readout Chips (ROC), each with 52×80 pixels of size 100×150 µm 2 . The ROC reads out the pixel data in double columns, with each double column having its own data and time-stamp buffer [1]. The structure of a ROC can be seen in figure 2. A more detailed description of the detector can be found in [2].

Dynamic Inefficiency
The efficiency of the pixel detector can be decreased by a number of different possible mechanisms. There are permanently damaged detector parts (many of them have been repaired or replaced for the next run period), modules with readout errors (buffer overflows, time-outs) and Single Event Upsets (SEU). The last one is caused by ionising radiation, which can cause the memory state of a logical element of the detector to flip. This may affect individual pixels, ROCs, or the readout electronics for entire modules. The SEUs can be fixed by reprogramming. Once these have been taken into account, there remains a significant efficiency loss that is known as the dynamic inefficiency. In a series of high multiplicity events, the buffers (mainly data and time-stamp buffers) of the ROC may overflow, resulting in data losses that decrease the efficiency. The inefficiency is dynamic  in the sense that the size of the effect depends on both previous and current events. This means the detector has a maximum efficiency just after the abort gap of the LHC. In the case of a buffer overflow, all the hits of the double column are lost. Individual pixels and entire ROCs can be inefficient, but data suggests that double column loss is the dominant effect.
We measure the performance of the detector in terms of hit efficiency [3], which is the probability to find a pixel cluster in any given sensor along a charged particle trajectory. The efficiency is measured taking into account any damaged modules and SEU candidates.

Simulation
In order to properly simulate dynamic inefficiency one would need to use the full simulation of the ROCs, in addition to storing the history of several events, neither of which is possible in the current CMS simulation software. Therefore, a data-driven method has been developed, in which the hit efficiency is parametrised for each module as a function of instantaneous luminosity and module position (layer, ladder and ring coordinates). This way, the dynamic inefficiency is independent of the quality of the physics simulation, but has to be calibrated for different running conditions. In the simulation double columns of pixel hits are randomly removed in proportion to the double column efficiency derived from data. In all three layers of the barrel part of the detector, the dynamic inefficiency is simulated in this way.
Module position is determined by ladder coordinates, ring coordinates and layer number. The ladder and ring coordinates are defined in the CMS global coordinate system: x points towards the centre of the LHC ring, y points upwards to the surface, and z points along the beam line. This can be seen in figure 1. The x = y = z = 0 is the interaction point in the centre of the CMS detector. The ladders are numbered along the r − φ plane. The coordinates' sign corresponds to the x axis sign; the numbers start from x = 0 and run from the +y side to the −y side. The rings are numbered along the beam axis with the interaction point in the middle. The coordinates' sign corresponds to the z axis sign; the numbers increase in magnitude moving outwards in ±z.
Results of the simulation are shown for layer 1 where the effect is most visible. The method has been validated by comparing simulations with, and without, dynamic inefficiencies to data. In figure 3 the dynamic inefficiency simulation agrees with the data by construction, as the double column efficiency in the simulation was set to reproduce hit efficiency in data. However it does not perfectly reproduce the data efficiency, as the double column efficiency is not the only factor affecting the hit efficiency. Ladder coordinates correspond to azimuthal angle φ , in which the detector is symmetric. However because the beam was not perfectly centered (beam offset) there is an azimuthal asymmetry in figure 3(a). In figure 3(b), the simulated hit efficiency measured in the rings numbered ±4 (corresponding to the rings at each end of the barrel region) do not agree with those observed in the data. This is due to the fact that the hit efficiency in that region is insensitive to double column loss. This subject is discussed in the next section. In figure 3(c) it can be seen that the dependence of the hit efficiency on the instantaneous luminosity is well reproduced in the simulation.

Results
The simulation of the dynamic inefficiency based on double column loss can be verified by studying the track incidence angles. The incidence angles are defined in the local coordinate system, which can be seen in figure 4(a). The double column direction points along global azimuthal angle φ , this can also be seen in figure 2. The incidence angle α points along the double column direction, whilst the incidence angle β corresponds to the global polar angle θ (or equivalently, the pseudorapidity, η) which is perpendicular to the double columns. Based on these definitions, we expect that if the dynamic inefficiency is caused by double column loss, the hit efficiency should be independent of α, but dependent on β . This trend is indeed observed in figure 4.
The incidence angle dependence of the hit efficiency supports the assumption that double column loss is the dominant effect. However, in figure 4(c) it can be seen that the shape of the distribution in the simulation does not agree with that observed in the data. For values of β close to 90 • degrees, an incoming (grazing) track creates a long cluster and the loss of a double column will only have a small effect, because β is perpendicular to the double column direction. A cluster "cut" in half by a double column loss is called a split cluster. Split clusters will still most likely be matched to the track, therefore the hit efficiency is unchanged. A track perpendicular to the module plane (β ≈ 0 • ) causes a small cluster which is more likely to lie on the boundary of a lost double column; as such the efficiency loss would be greater. The modules on ring numbers ±4 (the rings at each end of the barrel region) are likely to have tracks with incidence angle β close to 90 • .
In order to study the dynamic inefficiency many parameters have been examined; the distribution of the number of tracks and clusters showed the most improvement. The results can be seen in figure 5. The inefficiency of the forward disks is not simulated, however the effect of the barrel dynamic inefficiency simulation on the forward disk 1 can be seen in figure 5(a). Tracks crossing the forward region of the detector are likely to have hits from the barrel region as seeds for the tracking. Since these hits can be lost by dynamic inefficiency it affects the tracking in the forward region. The description of the distribution of the number of clusters has improved due to proper simulation of cluster splitting caused by double column loss.

Conclusion
The CMS simulation software has been improved by taking into account the dynamic inefficiency of the pixel detector barrel region. The advantage of using a data-driven method is that it is independent of the quality of the physics simulation. The disadvantage is that it needs to be calibrated using data from different running conditions. The technique has been validated by comparing several variables in data and simulation. The new simulation shows better agreement with data. Further improvement of the simulation is possible, for example, by including entire ROC or individual pixel loss and the extension of the efficiency loss to the forward disks.