Study of the Clocking Effect in the TRT Alignment

Transcription

1 Draft version 2. Study of the Clocking Effect in the TRT Alignment John Alison Aart Heijboer Joel Heinrich Joe Kroll University of Pennsylvania Andrea Bocci Duke University April 15, 28 Abstract This paper provides a brief discussion of the TRT alignment in the recent CSC alignment challange and describes several studies of the clocking effect, a p T dependent p T biasing, in the TRT alignment algorithm.

2 April 15, : 7 DRAFT 2 Contents 1 Introduction 3 2 Nominal Study 5 3 Summary of Other Studies Effect of Initial Misalignments Different Degrees of freedom Alignment by layer More Statistics/Iterations Global Vs Local χ 2 method Radial Comments Impact on Tracking 21 5 Conclusions 24 A Rotation Study 25 B Comparison of releases and 13.x. 27 C Study of the dependence of the alignment on radial misalignments 28

3 April 15, : 7 DRAFT 3 1 Introduction In 27 the Atlas community held the Computer System Commissioning (CSC) challenge [1], a project wide test of the software and computing infrastructure required by collisions data taking. Playing a central role in this challenge was the task of the inner detector community to test their alignment algorithms and calibration procedures on realistic Monte Carlo event samples. For this purpose, commissioning events were generated with initial misalignments thought to be true of a realistic detector installation. In the fall of 27 the alignment community released an initial set of alignment constants to the collaboration for use in other CSC analyzes, derived from these commissioning samples. This study is an attempt to clarify and understand discrepancies seen in these initial alignment constants with the input misalignments known a priori. In particular, it was seen by the alignment monitoring group that when the alignment constants found for the Transition Radiation Tracker (TRT) were included with those of the silicon tracking devices (Pixels and SCT) a systematic biasing of the transverse momentum(p T ) of reconstructed tracks with respect to the true p T of the simulated tracks was introduced. A possible explanation of the biasing seen is a misalignment effect known as clocking. This note describes a study of the clocking effect in the TRT alignment algorithm (TRTAlignAlg) [2]. In the CSC alignment challenge the alignment of the TRT was done with TRTAlignAlg, in two steps. First the TRT barrel was aligned internally, using tracks that were reconstructed using only TRT information. Then the TRT barrel and two endcaps were globally aligned to the rest of the Inner Detector, using tracks common to both the TRT and the silicon subsystems. Shortly after the first results of the alignment were released, validation of the alignment constants was preformed using the Atlas alignment monitoring package (InDetAlignmentMonitoring). [3] During the monitoring, it was then seen that when the TRT alignment was included a p T dependent biasing of track p T was introduced. When compared to the truth information negatively charged tracks tended to be reconstructed with higher p T, whereas positively charged tracks tended to be reconstructed with lower p T. This biasing can be seen in the blue lines in Figure 1, where the ratio of reconstructed p T to true p T is plotted as a function of the the true track p T, in the TRT barrel, using the alignment constants found for the TRT. As mentioned above, a possible explanation of this systematic bias is the clocking effect. This effect is part of a larger group of effects that arise when degrees of freedom in the detector alignment are unconstrained or weakly determined. These so called weakly determined modes are misalignments introduced which change the parametrization of the track from its correct helical representation to another, different helical representation by systematically altering the track parameters. In the case of clocking the detector misalignment is a relative rotation of detector layers with respect to others about an axis of symmetry, and the parameter that is biased is the track p T. As a result of how the TRT is aligned, internally first and then with respect to the rest of the inner detector, there are a few ways in which clocking can be introduced via the TRT. The first two are via internal or level 2 (L2) misalignments. The TRT barrel is composed of three layers of modules, each with full φ coverage and at increasing radius. A rotation of the outer module layers with respect to the inner layers would result in a clocking effect. In addition to a rotation of the layers, rotations at the individual module level can also give rise to a clocking effect. Each TRT layer is composed of 32 modules which are aligned individually in the alignment algorithm. It has been shown Appendix A, that a uniform rotation of all modules

4 April 15, : 7 DRAFT 4 Figure 1: reconstructed p T over true p T vs true p T for tracks with: no TRT information (black), TRT globally aligned and internally in ideal position (red) and, TRT aligned globally and internally (blue), about their center of gravity also leads to clocking. The final means by which the TRT may introduce the clocking effect is by an overall, global or level 1 (L1) rotation with respect to the silicon detectors. In this case the TRT may be perfectly aligned internally and give rise to clocking simply by a global misalignment (Appendix A). In principle a clocking type effect, systematically biasing track p T which is a due to the TRT can come from any one, or any combination, of these mechanisms. By referring back to Figure 1 we can get a handle on where the effect is appearing. Here the lines refer to tracks reconstructed after running the alignment on, only silicon with no TRT information(black), the silicon and the TRT at L1 while using the ideal L2 positions for the TRT (red), the silicon and the TRT aligned at L1 and L2 (blue). We conclude from the figure that little or no bias is introduced by a global misalignment of the TRT with respect to the rest of the inner detector, but rather by misalignments internal to the TRT. The remainder of this paper is devoted to an investigation of the internal L2 TRT alignment towards an understanding of the nature and origin of the p T biasing discussed above and is organized as follows. Section 2 and Section 3 present the results of several studies examining a possible clocking effect in the TRT alignment algorithm. Section 4 places the relevance of the TRT clocking effect in the context of tracking and track performance. Finally, Section 5 presents our conclusions and provides some questions that remain.

5 April 15, : 7 DRAFT 5 2 Nominal Study In order to definitively determine that the p T biasing seen in the CSC alignment is due to clocking in the TRT, and via which mechanism the effect is introduced, the TRT alignment as done in the context of the CSC challenge was repeated here with a few simplifications. First of all, because it was suspected that the clocking was due to TRT misalignments at L2, the alignment done here was only on the TRT at L2. The silicon detectors were placed in their ideal positions (ie: corrected for the initial misalignments) and the TRT was corrected for its global misalignment with respect to the silicon for simplicity. Secondly, because L2 misalignments of the TRT in the CSC Monte Carlo data are only included for the barrel, and because a significant bias of track p T from the TRT endcaps was not seen, only the TRT barrel was aligned in this study. Again with the two TRT endcaps in their ideal positions for simplicity. As mentioned above, when the alignment constants for the TRT were originally produced for the CSC challenge the TRT was aligned internally using tracks reconstructed with no silicon information. In order to determine if the clocking effect is dependent on this use of TRT only tracks or a more general feature of the algorithm or the TRT geometry, the internal alignment of the TRT was done with full, extended tracks. That is, the tracks used in the alignment contain both silicon and TRT information. As in the case for the CSC challenge the alignment was done with the TRT alignment algorithm (TRTAlignAlg) using the local χ 2 method and with events containing ten muons each. These multi-muon events were generated from distributions flat in φ, η, and p T (from 2 to 5 GeV), but for this study only tracks with an η of less than.8 were used for the alignment. The alignment was preformed in Athena release and iterated nine times 1) with 5 events each, minimizing five degrees of freedom per module (three rotations about the center of gravity and two translations perpendicular to the beam axis). The entire alignment procedure was then repeated with nine disjoint event samples 2). The initial misalignments of the TRT modules can be seen in Figure 2 where the initial displacements from the ideal, simulated in the CSC Monte Carlo, are shown for the three TRT layers in the upper and bottom left panels, as well as the projections of these layers into the ˆφ direction in the bottom right panel. It is in this bottom right panel in which the first type of clocking effect described above would manifest itself. Indeed, here we see that the initial CSC misplacements contain a clocking effect. [4] The arrows are color coded such that translations parallel to ˆφ in the clockwise direction are red, whereas translations in ˆφ in the counter clockwise direction appear blue. The predominance of red to blue arrows in this panel indicates that an initial clocking effect is present. Figure 3 displays the residual misalignments, after the nine alignment iterations for a typical event sample. We see that while on the whole the alignment algorithm converges to the correct positions, that is the module displacements decrease as they approach their correct values, a relic of the original clocking remains in the bottom right panel. Examining this panel we find that although the size of the initial clocking effect is reduced in magnitude the direction is preserved. It is also seen that most of the arrows that were previously misaligned in the counter clockwise direction are now in the clockwise direction. 1) The number of iterations is dictated by requiring the change in χ 2 per degree of freedom from the previous iteration to be below a certian threshold value, throughout this note that value has been taken to be 1 2) The joboptions used in the study described here can be found at /afs/cern.ch/user/j/johnda/public/clockingstudy/joboptionsnominalstudy.py

6 April 15, : 7 DRAFT 6 translation x 5 - Layer translation x5- y [mm] 1 y [mm] mm 1 mm x [mm] x [mm] translation x5 - Projection of the translation along #φ x 5 y [mm] 1 y [mm] mm.1 mm x [mm] x [mm] Figure 2: Initial TRT module displacements as simulated for the CSC alignment challenge. In order to quantify this effect pull distributions in ˆφ were made for each module and for all nine event samples. The pulls are defined as the displacement of the TRT module from the ideal position projected into ˆφ direction divided by the error on that position provided by the alignment algorithm, with a sign depending on direction of the displacement, clockwise being positive and counter clockwise negative. The results of all events samples after the alignment procedure can be seen in Figure 4, the upper left plot shows the pulls of all layers fitted with a Gaussian, and the pulls layer by layer are seen in the other three plots. Figure 5 shows the corresponding misalignments associated with the pulls in Figure 4. The pull distributions show means significantly different from zero, indicating that the alignment tends to result in a net displacement in clockwise ˆφ direction, which is identified as a clocking effect. The biasing of the pulls above zero, found when including the different event samples, suggests that the clocking seen in Figure 3 is not a result of a statistical fluctuation in that particular event sample, but a more robust phenomena. Although we see an effect that can

7 April 15, : 7 DRAFT 7 translation x 5 - Layer translation x5- y [mm] 1 y [mm] mm 1 mm x [mm] x [mm] translation x5 - Projection of the translation along #φ x 5 y [mm] 1 y [mm] mm.1 mm x [mm] x [mm] Figure 3: Residual TRT module displacements after alignment procedure from CSC initial displacements for a typical event sample. be associated with the p T biasing seen above, it remains unclear whether this is purely a function of the initial CSC misalignments, or something more closely associated with the algorithm itself. To gain further insight into this question a number of other studies were conducted, the results of which are presented in the next section.

8 April 15, : 7 DRAFT φ pulls for all layers Entries 864 Mean RMS 4.73 χ 2 / ndf / 33 Constant ± 2.61 Mean 4.78 ±.179 Sigma ± φ pulls for modules in layer Entries 288 Mean 2.32 RMS φ pulls for modules in layer 1 25 Entries 288 Mean RMS φ pulls for modules in layer 2 Entries 288 Mean RMS Figure 4: ˆφ pulls for TRT modules after alignment from CSC initial displacements. All layers are shown in the upper left, modules from layer are shown in the upper right, and layers 1 and 2 are shown on the bottom 1 8 φ residuals for all layers Entries 864 Mean.1625 RMS.1662 χ 2 / ndf / 27 Constant ± 4.4 Mean.1574 ±.54 Sigma.1497 ± φ residuals for modules in layer Entries 288 Mean.9139 RMS φ residuals for modules in layer 1 Entries 288 Mean φ residuals for modules in layer 2 Entries 288 Mean RMS RMS Figure 5: ˆφ misalignments for TRT modules after alignment from CSC initial displacements. All layers are shown in the upper left, modules from layer are shown in the upper right, and layers 1 and 2 are shown on the bottom

9 April 15, : 7 DRAFT 9 3 Summary of Other Studies In order to better understand the problems described in preceding section, we repeated the alignment procedure described there under various different conditions. The present section aims at collecting and presenting all these various studies, in hope of gaining further insight into how and where clocking arises. 3.1 Effect of Initial Misalignments To judge the effect of the initial displacements on the residual misalignments, the study described above in Section 2 was repeated with the detector in different initial positions. In the first such test that was done, the study presented in Section 2 was repeated with the same event samples and under the same conditions, however this time the TRT barrel was also placed in its ideal or perfect position, that is, the alignment algorithm was run starting with no initial inner detector misalignments. In order to quantitatively examine misalignments in ˆφ, pulls distributions for the TRT modules for all nine event samples were made in the same way described above and are shown in Figure 6, with the corresponding misalignments shown in Figure 7. Again we find means differing from zero with errors that are seemingly underestimated. Comparing these systematic shifts of the ˆφ pull distributions to those seen in Section 2, we can conclude two things. First, by comparing the magnitudes of the residual clocking found in the two cases we determine that initial misalignments do play a role in the determining the magnitude of the residual clocking. When starting with a detector which is initially misaligned and contains initial clocking we find a larger residual effect than when beginning in the ideal positions. Secondly, although the size of systematic shifts of the pull distributions are somewhat smaller than was seen in Section 2, the clocking effect is still introduced, and in the same direction, indicating the presence of a component of the clocking effect which is independent of the initial misalignments. As mentioned above, the CSC Monte Carlo used in these studies contain radial as well as azimuthal misalignments. To gain a deeper understanding of the nature of their role in determining the resulting clocking, the alignment was repeated with the initial CSC misalignments projected in both the ˆr and the ˆφ direction. In the first case the initial displacements are purely radial and thus no initial clocking is present. For the ˆφ misalignments the opposite is true, as the initial misalignments introduce the same amount of clocking as is inherent in the nominal CSC misalignments. The alignment procedure as described in Section 2 was then repeated on one of the event samples, using the two initial positions. The results, after nine iterations of the alignment algorithm, are collected in Figures 8 and 9 where the module pulls and misalignments along ˆφ of the resulting detector positions, beginning with different initial positions, are shown for the three layers separately. The circles indicate the mean of the distributions and the error bars represent the statistical uncertainty on those means. Figure 8 shows that when the initial alignments are radial the ˆφ pulls resemble those found when using the ideal starting positions, whereas the pulls found starting with the misalignments in ˆφ are similar to those when beginning with the nominal misalignments. From this it is concluded that only the initial misalignments in ˆφ direction, that is only the initial amount of clocking, plays a role in the determining the residual effect and whatever underlying mechanism is responsible for introducing the level of clocking seen when starting from the ideal position is

10 April 15, : 7 DRAFT φ pulls for all layers Entries 864 Mean RMS χ 2 / ndf 33.6 / 29 Constant 78.3 ± 3.52 Mean 2.55 ±.12 Sigma ± φ pulls for modules in layer Entries 288 Mean 1.35 RMS φ pulls for modules in layer 1 Entries 288 Mean 3.61 RMS φ pulls for modules in layer 2 25 Entries 288 Mean RMS Figure 6: ˆφ pulls for TRT modules after alignment from perfect initial positions. All layers are shown in the upper left, layer is shown in the upper right, and layers 1 and 2 are shown on the bottom also the dominating mechanism at work when beginning with only radial misalignments. The conclusion that the amount of residual clocking present after running the alignment algorithm is dependent on the amount of clocking in the initial misalignments was further studied, exploring the relationship in more detail. The alignment procedure as described in Section2 was again repeated 3) using various initial misalignments. In this case the misalignments used were multiples of the nominal CSC misalignment, projected in to the ˆφ direction. That is, the individual module misalignments used in the CSC data were scaled by a factor, between minus five and five, and then radial component of the resulting misalignment was removed. The initial misalignments were projected into ˆφ direction in light of the conclusion that only misalignments in this direction play a role in determining the residual amount of clocking. Figures 1 and 11 show the pulls in ˆφ and ˆr as a function of the scale factor used. The nonzero slopes of the ˆφ pulls seen in Figure 1 indicate the sensitivity of the alignment algorithm on the initial module positions. As the amount of clocking varies from five times the initial CSC clocking, through no clocking in the ideal positions, to five times the initial clocking in the opposite direction, the remnant effect decreases to near zero for all layers and then turns over resulting in clocking of an opposite sign than was seen above. The horizontal displacement of the three lines above the origin corresponds to the component of the effect independent of initial misalignments, seen in the clocking found when starting from the perfect positions. In contrast, the ˆr pulls seen in Figure 11 show an independence of a radial expansion or contraction on the initial ˆφ misalignment 4). 3) Here the alignment procedure was done in release 13.x., see Appendix B for a comparison of clocking found there to ) Appendix C repeats the study just described using radial, rather than ˆφ, misalignments

11 April 15, : 7 DRAFT φ residuals for all layers Entries 864 Mean.8942 RMS.1313 χ 2 / ndf / 22 Constant 11.1 ± 5.4 Mean.8693 ±.424 Sigma.126 ± φ residuals for modules in layer Entries 288 Mean.5864 RMS φ residuals for modules in layer 1 Entries 288 φ residuals for modules in layer 2 Entries Mean Mean RMS RMS Figure 7: ˆφ misalignments for TRT modules after alignment from perfect initial positions. All layers are shown in the upper left, modules from layer are shown in the upper right, and layers 1 and 2 are shown on the bottom 3.2 Different Degrees of freedom Along with the initial misalignments, another aspect of the alignment which may provide a clue as to how clocking is being introduced is the number of degrees of freedom being aligned. Throughout the studies described thus far each module has been aligned using five degrees of freedom. To determine if the clocking seen is due solely to a particular degree of freedom, or whether an interplay among them is responsible, the alignment was repeated using different variations on these degrees of freedom. Figures 12 and 13 presents the ˆφ pull distributions and the residual misalignments of several such studies. Each of these studies were done with the same event sample and using the alignment method of Section 2 starting with the nominal CSC misalignments, varying only the degrees of freedom aligned. The first bin shows the ˆφ pulls when all five degrees of freedom are fit. 5) Due to the geometry of the TRT, translations along the z axis of the modules are basically unconstrained and for similar reasons small rotations of the TRT about the global x and y-axes are also poorly determined by the alignment algorithm, but normally fitted for when doing the alignment at L2. It could be imagined that the extra freedom in the χ 2 fit provided by these two poorly constrained rotations can produce or enhance a possible clocking effect. However in the second bin, which shows the result of the alignment using only the well constrained degrees of freedom, translations in x and y and rotations around the z-axis, we find that the ˆφ pulls are essentially unchanged and conclude that the clocking seen here is independent of the poorly 5) The discrepancy in the magnitude of the mean of pull distributions shown in Figure 12 for all 5 dof and in Figure 8 for the nominal misalignments, is due to the discrepancy in Athena release used. The studies shown in this graph were done in 13.x., whereas those presented in Figure 8 were with See Appendix B

12 April 15, : 7 DRAFT 12 Mean φ Pull pull mean φ Layer CSC Nominal Misalignments Perfect Positions Only Radial Misalignments Only φ Misalignments Figure 8: Resulting ˆφ pulls after running the alignment with various initial misalignments constrained degrees of freedom. The appearance of the effect is further isolated by running the alignment using only translations. The third bin shows the results for this case. Here again, we find no major discrepancy with the ˆφ pulls seen for the alignment of five degrees of freedom. This result indicates that there is almost no sensitivity of the magnitude of the clocking effect to aligning rotations about the module z-axis. Appendix A shows that such rotations can indeed lead to a clocking effect, however from bin three of Figure 12 it is concluded that the effect studied in this note is not introduced through this mechanism. Finally, the alignment was done using only one degree of freedom, translations in ˆφ. The last bin shows the ˆφ pulls when starting with the modules constrained to be in their correct radial positions, because the alignment in ˆr is not being done, but using the same amount of misalignment in ˆφ, found the nominal case. The similarity, in the pull distributions when aligning all with five degrees of freedom to what is seen in this fourth bin, shows that almost all of the clocking seen above stems from the alignment (or lack thereof) of module translations in the ˆφ direction and that the effect of cooperation among other degrees of freedom plays a negligible role in introducing or enhancing the effect. 3.3 Alignment by layer In the previous section we saw that the clocking effect seen was due to module translations in ˆφ. In this section we examine another aspect of the TRT alignment through which this type of clocking effect can be studied, the correlations between the module layers. The clocking described thus far has been a collective effect of all three TRT module layers. Figures 14 and 15 show the ˆφ pulls and the residual misalignments of the modules resulting from aligning different combinations of the TRT barrel layers. The alignment was done beginning with the nominal misalignments for the modules in the layers being aligned, and the modules layers which were not aligned were restricted to their correct positions. The conclusion drawn from Figure 14 is that the clocking effect described in this note is

13 April 15, : 7 DRAFT 13 mean φ residual (mm) Mean φ residuals Layer CSC Nominal Misalignments Perfect Positions Only Radial Misalignments Only φ Misalignments Figure 9: Resulting ˆφ misalignments after running the alignment with various initial misalignments indeed a collaborative effect among all three layers. When the layers are aligned individually, the resulting misalignments in ˆφ are much smaller than that seen when aligned in unison. When aligned in pairs the implication of the ˆφ pull distributions is not as obvious. The explanation of the pulls given here is that the clocking effect tends to be larger when aligned in pairs, however the restriction of the third layer to be in the ideal position has an anchoring effect on the resulting clocking. When only the first two layers are aligned the outer module layer and the silicon detectors serve as fixed points, inhibiting clocking. Similarly when inner and outer module layers are aligned, the fixed middle layer serves as a barrier to conspiring misalignments which could preserve the helical form of the fitted tracks, and results in clocking similar to that seen when aligning the layers individually. However, in the case when the outer module layers are aligned there is nothing fixing the module positions at larger radii or prohibiting collaboration of the two modules, giving rise to a larger effect than when aligned individually. Adding the additional inner module layer further enhances the effect, bringing the magnitude clocking effect up to which was has been presented above. 3.4 More Statistics/Iterations All the studies described thus far have involved the same basic alignment procedure, using 5 multi-muon events and iterating the alignment algorithm nine times. Figures 16 and 17 show the ˆφ pulls and residual misalignments when more statistics or iterations are used. The first two bins give the results of running the alignment from the nominal and ideal positions for nine iterations, whereas the following two bins show the results using twice the iterations. When starting from the nominal misalignment doubling the number of iterations reduces the residual clocking effect to near the level seen when starting from the ideal case. However increasing the number of iterations has no effect on the resulting clocking when the alignment is begun in the perfect positions. The final two bins show the result of increasing the number of events used in the alignment procedure by a factor of ten, to 5. When using higher statistics the

14 April 15, : 7 DRAFT 14 Mean φ Pull vs multiple of CSC (initial misalignments projected in the φ direction) mean φ pull Layer xcsc Figure 1: Mean of the ˆφ pull distribution vs CSC scale factor. (Misalignment in ˆφ only) pull distributions are much larger than was seen before, however from looking at the module misalignments in Figure 17 we find that the actual displacements from the true positions are smaller and can conclude that the high shifts seen in the ˆφ pull distributions are driven by the decrease in the errors. 3.5 Global Vs Local χ 2 method The TRT alignment algorithm TRTAlignAlg computes the alignment constants by minimizing the χ 2 function with respect to the parameters being aligned [2]. TRTAlignAlg supports two ways of doing this, the local and global χ 2 method. In the global method the minimum χ 2 condition is linearized, first and second derivatives of the χ 2 are calculated with respect to an initial alignment position, and the second derivative matrix is inverted. The local χ 2 approach simplifies the problem by ignoring correlations between modules, thereby having only to invert much smaller matrices. However because the detector elements are not uncorrelated, iterations are necessary in the local approach to solve for the minimum. The studies described thus far were done using the local χ 2 method. From Figures 18 and 19 which compare the two approaches, we can see that running the global method, beginning in the nominal positions improves the resulting clocking to the level seen when starting from the ideal positions. Alternatively, if the global method is run from the ideal positions the results are similar to those found when using the local method.

15 April 15, : 7 DRAFT 15 Mean R Pull vs multiple of CSC (initial misalignments projected in the φ direction) mean r pull Layer xcsc Figure 11: Mean of the radial pull distribution vs CSC scale factor. (Misalignment in ˆφ only) 3.6 Radial Comments Throughout this section, and indeed the entire paper, we have been concerned primarily with the module pulls in ˆφ and have paid little or no attention to the pull distributions in the radial direction, however referring to Figure 2 we can see that they are fairly insensitive to changes made to the alignment procedure discussed in this section. It seems that on average the alignment algorithm tends to move the modules out radially with almost no variation due to the initial misalignment or the degrees of freedom involved.

16 April 15, : 7 DRAFT 16 Mean φ Pull pull mean φ 12 1 Layer All five Dof Only well constrained Dof Only translations) Only translations in φ Figure 12: Resulting ˆφ pulls after running the alignment with various degrees of freedom Mean φ residuals residual (mm) mean φ Layer.2.1 All five Dof Only well constrained Dof Only translations) Only translations in φ Figure 13: Resulting ˆφ misalignments after running the alignment with various degrees of freedom

17 April 15, : 7 DRAFT 17 Mean φ Pull pull mean φ 1 8 Layer All three layers Layer Layers and 1 Layers 1 and 2 Layers and 2 Figure 14: Resulting ˆφ pulls after aligning different combinations of the module layers Mean φ residual residual (mm).35.3 Layer mean φ All three layers Layer Layers and 1 Layers 1 and 2 Layers and 2 Figure 15: Resulting ˆφ misalignments after aligning different combinations of the module layers

18 April 15, : 7 DRAFT 18 Mean φ Pull pull mean φ 2 15 Layer Iterations (Nominal) 9 Iterations (Perfect) 18 Iterations (Nominal) 18 Iterations (Perfect) High statistics (Nominal) High statistics (Perfect) Figure 16: Resulting ˆφ pulls after aligning for more iterations and with more statistics Mean φ residual residual (mm) mean φ Layer Iterations (Nominal) 9 Iterations (Perfect) 18 Iterations (Nominal) 18 Iterations (Perfect) High statistics (Nominal) High statistics (Perfect) Figure 17: Resulting ˆφ misalignments after aligning for more iterations and with more statistics

19 April 15, : 7 DRAFT 19 Mean φ Pull pull mean φ 12 1 Layer Local method (Nominal) Local method (Perfect) Global method (Nominal) Global method (Perfect) Figure 18: Comparison of the ˆφ pulls after aligning using the local and global χ 2 methods residual (mm) mean φ Mean φ residual Layer Local method (Nominal) Local method (Perfect) Global method (Nominal) Global method (Perfect) Figure 19: Comparison of the ˆφ misalignments after aligning using the local and global χ 2 methods

20 April 15, : 7 DRAFT 2 Mean R Pull mean r pull 5 4 Layer Nominal Misalignments(No Rotations) Nominal Misalignments(No Unconstrained Dof) Only φ Misalignments Only Radial Misalignments 13.x. Perfect Positions 13.x. CSC Nominal misalignments 13..x Perfect Positions 13..x CSC Nominal misalignments Nominal Misalignments (Only Align in φ) Only φ Misalignments (Only Align φ) Perfect Positions (Only Align φ) Figure 2: Survey of Clocking Study: ˆr pulls

21 April 15, : 7 DRAFT 21 4 Impact on Tracking In order to address the significance of the size of the clocking effect seen in the studies described above, this section examines their impact on tracking. Apart from Figure 1, another way in which the clocking effect manifests itself is through its impact on the track p T pull distributions. Clocking biases positively and negatively charged tracks in opposite ways, causing their pull distributions to shift apart or separate. Figure 21, shows the p T pull distributions for tracks reconstructed with the ideal geometry(upper left), using the residual misalignments resulting from aligning from the perfect positions (upper right), and using the residual misalignments of nominal study as shown in Figure 3(lower left). The p T pull distributions shown are defined as the difference between the true Monte Carlo p T and the reconstructed p T, divided by the error associated with the reconstructed p T. The plots each contain the same 5 events and have tracks with an η of less than.5, insuring that they are reconstructed in the TRT barrel. Pulls for all tracks(black), and for positive(blue) and negative(red) tracks individually, are shown for each detector geometry. Indeed, we find in Figure 21 the separation of the means of the pulls for positive and negative tracks growing from.27 in the ideal case, to.277 after aligning from the perfect situation, further still to.494 when beginning the alignment with a realistically misaligned detector. 3 Pt Pulls (Ideal) Entries 9245 Mean.4831 RMS Pt Pulls (After Alignment From Perfect) Entries 9245 Mean.3982 RMS Entries Mean.3492 RMS Entries 466 Mean.618 RMS 1.3 Entries Mean RMS Entries 466 Mean.1787 RMS Pt Pulls (After Alignment From Nominal) 3 Entries 9245 Mean RMS Entries Mean RMS Entries 466 Mean.2916 RMS Figure 21: p T pulls for: all tracks(black), positive tracks(blue) and negative tracks(red) are shown reconstructed with various detector geometries As was mentioned above and can be seen in Figure 1, the impact of the clocking effect on the momentum bias varies with the track p T. What makes the clocking effect particularly

22 April 15, : 7 DRAFT 22 bothersome is that the biasing is worse in tracks with large p T, the ones most interesting from a physics point of view. To see this born out in our studies we have reproduced the track p T pull distributions in Figure 21 for tracks at the higher and lower end of our spectrum. Figure 22 show the pulls distributions for tracks with p T from 2 to 2 GeV on the left hand side and for tracks with 4 to 5 GeV on the right. The upper two plots are for tracks reconstructed with the perfect detector geometry, followed by tracks found using the residual misalignment in the perfect case, and finally the bottom two plots show the pull distributions for the high and low p T tracks using the detector after the alignment is run with the CSC misalignments. As advertised the separation between tracks with opposite charge grows with their p T, providing further evidence of clocking. The commissioning samples used throughout the studies presented in this paper contain tracks with p T reaching up to 5 GeV, but presumably in similar distributions produced for tracks with higher p T the impact of the effect would be seen to grow further still. At the level of the misalignments seen in the studies of Section 2 and 3 we see a clear impact on the reconstructed track p T s and thus in the physics in which one is interested.

23 April 15, : 7 DRAFT Low Pt Pulls (Ideal) Entries 349 Mean.721 RMS High Pt Pulls (Ideal) Entries 1928 Mean RMS Entries Mean.2336 RMS Entries 1741 Mean.121 RMS Entries 966 Mean RMS Entries 962 Mean RMS Low Pt Pulls (After Alignment From Perfect) Entries 3488 Mean.594 RMS High Pt Pulls (After Alignment From Perfect) Entries 1943 Mean RMS Entries Mean RMS Entries 1742 Mean.1734 RMS Entries 993 Mean RMS Entries 95 Mean.153 RMS Low Pt Pulls (Nominal) Entries 3488 Mean.664 RMS High Pt Pulls (Nominal) Entries 1938 Mean RMS Entries Mean RMS Entries 1742 Mean.2342 RMS Entries 11 2 Mean RMS Entries 937 Mean.3215 RMS Figure 22: p T pulls for: all tracks(black), positive tracks(blue) and negative tracks(red) are shown reconstructed with various detector geometries and for high and low p T seperately

24 April 15, : 7 DRAFT 24 5 Conclusions The internal TRT alignments reached in the recent CSC challenge were shown to give rise to a p T dependent p T biasing. In this note we have seen that this biasing is brought about by systematic misalignments of modules in the TRT barrel. It was seen that the effect is enhanced by the collaborative effort among modules in different layers and is insensitive to including other degrees of freedom in the alignment. Both of the TRTAlignAlg s methods of χ 2 minimization (local and global) were studied and shown to result in residual clocking when the alignment was run beginning with an ideal inner detector geometry. This suggests the presence of an aspect of the TRT alignment problem which is dependent on the inherent detector geometry or event topology which leads to clocking. Throughout this paper we have run the alignment using the highly idealized multi-muon event samples with negligible noise or background. The clocking effect seen in the TRT alignment in these simple situations must be viewed as a lower limit to what we can expect using more realistic event samples. If improvements are to be made they must come from exploiting fundamentally different event topologies or imposing other independent constraints on the alignment problem. Can cosmic events from recent and future milestone runs control or eliminate this effect? If so how many tracks are needed to bring the clocking down to an acceptable level? The TRT plays a crucial role in determining the momenta of tracks in the Inner Detector. In this note we have attempted to convey how misalignment, both internal to the TRT and relative to other sub-detectors, can disrupt this determination, and have hopefully provided some insight into how and why this can happen.

25 April 15, : 7 DRAFT 25 A Rotation Study As mentioned in Section 1 there are a number of different sources of the clocking effect in the TRT, appearing both as a result of global misalignments and due to misalignments internal to the detector. In this appendix we examine two of these mechanisms, a L1 rotation of the entire TRT with respect to the rest of the inner detector and rotations of the individual phi modules about their center of gravity, in order to determine the size of the induced effect as a function of the corresponding misalignments. Although the entire Inner Detector is insensitive to a global rotation about the beam axis, a relative angle between subdetectors does have a noticeable effect. In particular a rotation of the TRT barrel with respect to the other silicon detectors can result in a systematic biasing of track p T, the clocking effect. To study the magnitude of this effect the track reconstruction was preformed with the TRT barrel misaligned with various such a rotations. The results can be seen in Figure 23, where the mean of the p T pull distributions for negative and positive tracks are plotted separately, as a function of the magnitude of the level 1 rotation given to the TRT. The tracks in the pull distributions shown here were required to go through the TRT barrel, and are from 5 of the CSC muon events described in the main text. Pull Mean of the p T Mean p T Pull vs L1 rotation Positive Tracks Negative Tracks TRT L1 Rotation (µrad) Figure 23: means of the p T pull distributions for positively and negatively charged tracks are shown versus angle of L1 TRT rotation, In this context, the size of the clocking effect is identified with the splitting of the means of the positive and negative tracks pull distributions. Here we see that a L1 rotation introduces clocking and is, in fact, quite sensitive to small angles. With a misalignment of only 8 µrad, the separation between positive and negative tracks is already of order one σ. Although not addressed further in this paper, the clocking effect is seen to be sensitive to a small relative angular misalignment of the TRT and silicon trackers and will continue to pose a significant challenge to eliminating p T biases in future alignments procedures. Another means in which the clocking effect is expected to manifest itself is through rotations

26 April 15, : 7 DRAFT 26 of the TRT modules themselves. To study clocking under this guise, the TRT was placed in its ideal position with respect to the silicon, and then varying rotations were given the individual modules that compose the TRT Barrel. The modules were all rotated by the same angle and about the axis parallel to the beam pipe and through their center of gravity. Pull Mean of the p T Mean p T Pull vs L2 rotation Positive Tracks Negative Tracks Module (L2) Rotation (µrad) Figure 24: means of the p T pull distributions for positively and negatively charged tracks are shown versus angle of L2 rotation of TRT modules, Figure 24 shows results very similar to those seen in the case of the Level 1 rotations just discussed. Here again the clocking effect is seen, and with a large dependence on the Level 2 rotation angle. The similarity of the magnitude of the effect in these two cases along with the small size of the angles involved, leads us to conclude that to a close approximation that the actual physical movements of the straws associated with the modules is the same for both types of rotations.

27 April 15, : 7 DRAFT 27 B Comparison of releases and 13.x. Mean φ Pull pull mean φ Layer x CSC Nominal Misalignments 13..x Perfect Positions 13.x. CSC Nominal Misalignments 13.x. Perfect Positions Figure 25: Survey of Clocking Studies: ˆφ pulls. Clocking in the TRT was first studied by the present authors in Athena release Shortly after release 13..3, significant changes to the tracking regarding TRT information were implemented in Athena versions 13.x. and above. These changes included updating the errors associated to TRT hits, improving how ambiguities in tracks found in the TRT are resolved, and allowing the left-right assignment of the TRT driftcircles to occur at a later stage in the tracking. To determine what effect these changes had on the TRT alignment, the studies described in Sections 2 and in the begining of section 3, were repeated on the same events using the two different releases. Again, the alignment was done with 5 CSC multimuon events and for nine iterations, aligning five degrees of freedom. The results are shown in Figure 25. The first two bins in the figure show the pulls of the resulting module positions after aligning using release and starting from the nominal CSC misalignments(figure 2) and the perfect positions, respectively. The following two bins show the results using the updated tracking in Athena release 13.x.. Although the improvements made to the tracking seem to reduce the ˆφ pulls, and hence the resulting clocking, the effect as seen in is still present with clocking significantly different from zero.

28 April 15, : 7 DRAFT 28 C Study of the dependence of the alignment on radial misalignments The impact of the initial misalignments on the residual clocking effect was explored in Section 3. There the initial misalignments used were multiples of the nominal CSC misalignments projected in the ˆφ direction. This appendix presents the results of a similar study using instead the initial misalignments projected in the radial direction. The results are summarized in Figures 26 and 27. The means of the pull distributions of the ˆφ and ˆr misalignments are shown as a function of the scale factor used in the initial misalignments. In contrast to the previous findings, using the ˆφ misalignments, in the figures we see almost no dependence of the resulting misalignments on the initial radial positions of the modules. This provides further support to our conclusion that the residual clocking effect is only sensitive to initial misalignments in ˆφ. It may have been suspected that the residual misalignments in a particular direction are only sensitive to initial misalignments in that direction, as was the case in Section 3. However we see here that along with the misalignments in ˆφ, the ˆr module positions have little reliance on the initial misalignments. Mean φ Pull vs multiple of CSC (initial misalignments projected in the radial direction) pull mean φ Layer xcsc Figure 26: Mean ˆφ pull vs CSC scale factor.

29 April 15, : 7 DRAFT 29 Mean R Pull vs multiple of CSC (initial misalignments projected in the radial direction) mean r pull Layer xcsc Figure 27: Mean ˆr pull vs CSC scale factor.

30 April 15, : 7 DRAFT 3 References [1] D. Barberis, ATLAS plans for 26:Computing System Commissioning and Service Challenge 4, 26, hep.phys.sfu.ca/~rwalker/atlas_sc4_mumbai.pdf. [2] A. Bocci and W. Hulsbergen, TRT Alignment for SR1 Cosmics and Beyond, 27, atlindet-pub [3] B. Cooper, T. Golling, B. Heinemann, S. Strandberg, and J. Alison, First look at Pavel + TRT Alignment, IDAlignment/318.html. [4] A. Bocci and C. Schmitt, TRT Alignment, 27, contributiondisplay.py?contribid=13&sessionid=14&confid=9662.