Alignment of the ATLAS Inner Detector Tracking System
The ATLAS detector, built at one of the interaction points of the Large Hadron Collider, is operational and has been collecting data from cosmic rays. This paper describes the track based alignment of the ATLAS Inner Detector tracker which was performed using cosmic rays collected in 2008. The alignment algorithms are described and the performance of the alignment is demonstrated by showing the resulting hit residuals and comparing track parameters of upper and lower segments of tracks. The impact of the alignment on physics measurements is discussed.
The ATLAS detector ATLAS is a general purpose detector built at one of the interaction points of the Large Hadron Collider (LHC) where proton on proton collisions with a center of mass energy of 14 TeV are expected. The inner tracking system of ATLAS is made up of silicon detectors and straw drift tubes. While these detectors were placed with very high precision of the order of 100 , the precision required for physics necessitates determining the positions of the tracking elements to a few microns. This is only achievable by doing a track based alignment. The LHC has not yet started proton-proton collisions, however, the ATLAS detector is fully operational and has been collecting cosmic ray data. This paper describes the alignment achieved using this cosmic ray data and explores the expected impact of the alignment on the physics performance in the early data.
Ii Overview of the ATLAS Inner Detector
The ATLAS detector consists of several systems. An inner tracker (Inner Detector), electromagnetic and hadronic calorimeters and a muon spectrometer. The Inner Detector is located within a solenoidal magnetic field of about 2 Tesla and is made up of three subsystems as shown in Figure 1.
The innermost subsystem is the pixel detector. It is made up of three barrel layers and three disks in each endcap giving at least 3 space points per track. There are a total of 1744 modules (1456 in the barrel and 144 in each endcap). The pixel cell size is 50 400 with a corresponding resolution of 10 115 . The more precise measurement is in the direction (in the bending direction of the magnetic field) and the less precise direction measures in the barrel and in the endcap. In the local frame of the module the directions are referred to as local and local respectively.
The next subsystem is the SCT (Semi-Conductor Tracker) which consists of silicon microstrip detectors. In the SCT there are four barrel layers and 9 disks in each endcap giving 4 space points per track. It has a total of 4088 modules (2112 in the barrel and 988 in each endcap). The strip pitch is about 80 giving a resolution of 17 in the (local ) measurement direction. The modules are made of two back to back sides which are rotated 40 mrad with respect to each other to give a stereo measurement. This results in a space point resolution of about 580 in (barrel) or (endcap).
The outer subsystem is the TRT (Transition Radiation Tracker). It is made up of straw drift tubes which have a diameter of 4 mm. The straws are embedded in a material that produces transition radiation photons which facilitates electron identification. On average 36 straws are crossed per track. The barrel is segmented into 96 modules arranged in three rings. Each endcap is made up of 20 wheels, where each wheel consists of two four-plane structures. The resolution is 130 in the measurement direction.
Iii Cosmic Ray Data Collection
The data used to obtain the alignment presented here was collected from September to December 2008 using cosmic rays. In total 2.6 million Inner Detector tracks were recorded with the magnetic field on and 5 million with the magnetic field off. A number of these tracks, however, do not pass through all three subsystems. Requiring at least one hit in the SCT, results in 880K and 2 million tracks with field on and off respectively. Requiring at least one hit in the pixel detector, results in 190K tracks with field on and 230K tracks with field off.
Iv The Alignment Algorithms
The hit residual is the distance between the track prediction and the measured hit. The basic approach to align the detector is to reduce the residuals. The main methods build a that is to be minimized. This is defined as
where is a vector of the residuals and is the corresponding covariance matrix. The minimum is obtained by solving the condition:
where is a vector containing the alignment constants. These are generally the 3 translations and 3 rotations of each alignable structure (e.g. a module or layer). The solution to the minimization is of the form:
where a full derivation can be found in Alignment . The matrix is a matrix where is the number of degrees of freedom. If aligning all modules this will be 6 (three translation and three rotations) times the number of modules. In the case of the silicon detectors this results in a 35K 35K matrix. The solution of this large system of linear equations can be obtained by full diagonalization (for example with LAPACK LAPACK or ScaLAPACK ScaLAPACK ) which is computationally intensive or by fast solving techniques (e.g. MA27 MA27 ) which can be performed on a standard workstation. The fast solving methods rely on the matrix being sparse which is generally the case.
Solving this large matrix is referred to as the global approach Alignment ; AlignmentTRT . The correction of one module will be correlated to the movement of other modules and these correlations are taken into account in one go when solving this matrix. A few iterations are generally required due to non linearities.
A second approach is the local approach AlignmentTRT ; AlignmentLocal1 ; AlignmentLocal2 which ignores the correlations between modules and so one only needs to invert a matrix for each module. Correlations are taken into account by iterating several times.
The global is currently the baseline approach but both approaches are implemented in ATLAS and give consistent results.
In addition to the minimization techniques, a robust alignment algorithm AlignmentRobust has also been developed. This works by shifting modules according to their observed average residual offsets in an iterative fashion. In particular it takes advantage of the regions where modules overlap.
V The Alignment Strategy
The alignment sequence was as follows. First the silicon detector (both pixel and SCT together) was aligned internally. A more detailed sequence of the silicon alignment is described below. For the pixel detector, survey information was available and was used as a starting point for the alignment. Next the TRT was aligned internally. After that, the TRT was aligned with respect to the silicon detectors. Finally a “Center-of-Gravity” correction was made which adjusts the overall translation and rotation of the entire Inner Detector such that the aligned detector has the same center of gravity as the nominal detector. This is needed as the minimization is insensitive to the overall translation and rotation of the Inner Detector.
The alignment of the detector was done in a number of steps at different levels closely following the structural assembly of the detector. The placement of the larger structures are less precisely known than that of the precision of the assembly of the modules within their substructures. The first level of alignment (referred to as Level 1) was at the level of major subsystems which were installed as separate items. These are the 2 TRT endcaps and the TRT barrel, the 2 SCT endcaps and the SCT barrel and the whole pixel system (however, for the silicon internal alignment the TRT was not included). The next level of alignment was at the disk and layer level. Each of the three pixel barrel layers were constructed from two semi-circular half shells. At this alignment level the 6 pixel layer half shells plus the 4 SCT barrel layers were aligned. Due to the poor illumination of the endcaps (since most cosmic ray muons travel predominantly in the vertical direction) the disks within the endcap were not aligned separately but rather the endcaps were aligned as a whole (2 SCT and 2 pixel endcaps).
Finally a module level alignment was done. Before this module level alignment was made, it was observed that the pixel staves have a significant bow lateral to the module plane which was also expected from the mechanical construction of the detector. To remain conservative not all degrees of freedom were aligned at the module level but rather they were restricted to the two degrees of freedom which were able to correct for the stave bow. These degrees of freedom were the angle about the normal of the module and the translation in the precision measurement direction (local ). At the module level only barrel modules were aligned, the endcaps were again kept as a whole.
Since the alignment works by minimizing residuals, the distributions of the residuals are a key test of the performance of the alignment. Figures 2 - 5 show the residuals for the different detectors. The quoted in the figures is the of the Gaussian describing the core after doing a fit to a double Gaussian distribution. In the case of the pixel, both the high precision measurement direction (local ) and the direction orthogonal to this (local ) are shown. The figures show the distributions before alignment where one sees wide distributions which are not centered around zero. After the alignment the residual widths are reduced significantly and well centered on zero. The widths are approaching that of an ideal alignment.
If one takes the quadratic difference one estimates that the remaining misalignment is equivalent to a random displacement of the modules less than 20 . This was further tested by generating a residual misalignment set (labeled as “Day 1”) with modules misplaced randomly with Gaussian width of 20 in the local and directions which approximately reproduces what is seen in data as shown in Figure 6. It is expected, however, that the remaining misalignment is not just random misplacement, but quite likely also includes systematic distortions. This is discussed further in Section VII.2.
Data from cosmic ray runs in 2009 were also recently processed using the alignments obtained with the 2008 data. While the residual widths were slightly increased the mean positions were still well centered on zero indicating that the detector has been quite stable over an extended period of time.
vi.2 Upper and Lower Track Comparison
The reduction in width and the centering of residual is a necessary condition to demonstrate a good alignment. However, it is not sufficient as a number of systematic distortions are insensitive or weakly constrained by the minimization of the . Cosmic ray tracks have the unique feature that many cross the upper and lower parts of the detector. One can take tracks that pass close to the origin (i.e. where beam particles collide in collision events) and split the track into a lower and upper segment and then refit these as two independent tracks. It is possible then to compare the track parameters. This was done for tracks with GeV, mm and mm, where is the transverse impact parameter measured with respect to the origin and is the position at the point of closest approach to the origin. This cut ensures that the track has at least gone through the first pixel layer.
The result of this procedure for the impact parameter, where one looks at the difference of the impact parameter of the two tracks, is shown in Figure 7. Before alignment there is a large shift away from zero and a very broad distribution. After alignment the width is significantly reduced. A shift of 11 is however observed, indicating there is still some improvement needed which is still under investigation.
Since there are two tracks, an estimate of the impact parameter resolution can be obtained from the width of this distribution divided by . This result in a resolution of 35 . For comparison, the impact parameter resolution from collision events as estimated from Monte-Carlo is 20 for tracks with GeV.
Figure 8 shows the azimuthal angle, , of the track and again good improvement is seen after alignment and the distribution is well centered around zero.
Figure 9 shows the resolution of (the charge over momentum) as a function of the transverse momentum, , as obtained with this track splitting method. Both tracks reconstructed with the full Inner Detector (i.e. including the TRT) and those reconstructed with only the silicon detector are shown. The TRT, due to its large lever arm, is seen to significantly improve the resolution, especially at large momentum. Comparing tracks reconstructed with the full Inner Detector using perfectly aligned Monte-Carlo and tracks reconstructed in data, one can see that at low momentum the agreement is very good. At low momentum multiple scattering dominates and as expected the alignment is not as crucial for these tracks. However, for high momentum tracks there is less multiple scattering and the impact of the remaining residual misalignment can be seen.
Vii Prospects for Collision Data Taking
vii.1 Expectations for First Collision Data
As was mentioned above, the residual distributions from the cosmic ray data can be reproduced in Monte-Carlo by introducing random residual misalignments of the order of 20 (see Figure 6). The random residual misalignment set used in simulation was referred to as “Day 1” being an estimate of what is achievable on the first day of collisions given a starting alignment obtained solely from the cosmic ray data. While it is expected that we will rapidly improve the alignment with collision data, it is interesting to investigate the effects that this “Day 1” alignment has on physics observables. Therefore some physics channels were simulated with this misalignment set. In addition, a set was produced with a smaller amount of misalignment as an estimate of the alignment that might be achievable after a few months of running on collision data and is labeled as “Day 100” alignment. It should be cautioned, however, that there are many uncertainties to what will actually be achieved on this timescale. Table 1 shows the size of misalignment included in these two sets.
|Day 1||Day 100|
The impact on the mass resolution from events reconstructed using only the Inner Detector with the “Day 1” and “Day 100” alignment sets is shown in Figure 10 AlignmentAndPhysics . As can be seen in the figure the impact of misalignment in the “Day 1” scenario shows large degradation with a contribution to the mass resolution (in quadrature) of 2.2 GeV. For the “Day 100” set there is still some degradation, however it is much reduced with a contribution from misalignment of about 1 GeV.
The impact on observables of interest to physics was also investigated AlignmentAndPhysics by studying the impact on the mass resolution of (see Figure 11) and in events where and . While there is some degradation in the “Day 1” sample, the dominance of the lower tracks in these samples results in more of a contribution from multiple scattering and less from misalignment. The effects of misalignment from the “Day 100” set are reduced even further to almost an insignificant amount.
A study of the impact of misalignment on -tagging was made with alignment sets different from those described above. The results of this study CSCBook are summarized in Figure 12 which shows the rejection of light jets (jets originating from gluons, up, down and strange quarks) for a fixed -tagging efficiency. The light jet rejection is defined as the inverse of the efficiency of a light jet being tagged as a -jet. Different -tagging algorithms were used. The IP2D algorithm is based on the transverse impact parameter significance (the impact parameter divided by its error). The IP3D algorithm combines transverse impact parameter significance with the longitudinal impact parameter significance. The SV1 algorithm uses properties of secondary vertices such as the vertex mass and the number of tracks used to make the vertex. The IP3D-SV1 combines the information from the IP3D and SV1 taggers. In this study four alignment sets were considered. For the first set, Random10, modules were displaced and rotated randomly with a distribution with width of about 10 . Layers and disks and the whole pixel layer were also displaced by a similar amount. Only misalignment in the pixel subsystem was considered. Taking into account the misplacements at several levels, the overall misalignment is considered to be comparable to the “Day 1” set above. The second set, Random5, introduces misalignment about half the size. The third set is the case of a perfectly aligned detector. The forth set, Aligned, was produced using a simulation where large misalignments were introduced which were typical of what is expected when building the detector and then running the actual ATLAS alignment algorithms with a mixture of collision-like Monte-Carlo and cosmic ray Monte-Carlo. For the Random10 set the degradation for the IP3D-SV1 algorithm is significant (about 50%) but still one is able to achieve reasonable rejection rates. For the more realistic alignment case (Aligned) the degradation is only marginal with a 15% degradation. The impact parameter based taggers appear to be more sensitive to misalignment than the secondary vertex based taggers.
vii.2 Systematic Distortions
While random misalignments cause some degradation to the performance of physics, it is also likely there will be systematic distortions which could be difficult for the alignment to completely eliminate. Any deformation which will still allow helical tracks to be reconstructed will leave residuals unchanged. The minimization of the is not sensitive to such deformations. These deformations are also referred to as weak modes. One such deformation is a twist of the detector where the detector is systematically rotated as a function of the global coordinate. Another such deformation is a curl deformation where each subsequent layer is rotated progressively by larger amounts as a function of the radius. This particular deformation creates a momentum bias. Tracks with one charge get a larger momentum and tracks with the opposite charge get a smaller momentum. Such a deformation was introduced into an alignment set and the results on the mass reconstruction of in events was investigated. The size of the curl introduced was such that the outer most silicon layer was shifted around 300 . This set is labeled as “Curl Large”. In addition, the ATLAS alignment algorithms were run starting with a geometry with this curl misalignment. It was seen that the alignment procedures were able to remove much of the deformation indicating that the deformation was not a perfect weak mode. The alignment set obtained after running the alignment is labeled as “Curl Small”. The “Curl Large” is seen to give both a degradation in the resolution and introduces a bias in the mass. The “Curl Small” shows improvement but still some degradation with respect to the perfect alignment. The mass bias however is mostly removed. When running the alignment to produce the ‘Curl Small’ set, only collision-like Monte-Carlo was used. The addition of events from cosmic rays is expected to improve the alignment further.
The ATLAS Inner Detector has successfully taken a large amount of cosmic ray
data which has allowed a first full Inner Detector alignment to be achieved using the standard
ATLAS alignment algorithms. The residuals
obtained after alignment show much reduced widths and are well
centered on zero. The resulting alignment is equivalent to less than a 20
residual misalignment in the barrel. The results from comparing track
parameters from upper and lower segments also show good performance. The
alignment is already at a level where it is possible to analyse low
physics channels such as
those of interest for physics and the alignment is expected to rapidly improve
with collision data. The tackling of systematic deformation is still expected to be a
challenge although combining collision data with cosmic ray data is expected to help.
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