1. Geant4 Tracking Test (D. Lunesu)1 Daniela Lunesu, Stefano Magni Dario Menasce INFN Milano GEANT4 TRACING TESTs

2. Geant4 Tracking Test (D. Lunesu)2 The process of tracing a charged particle in a magnetic field is controlled, in Geant4, by several user-adjustable parameters. This study aims at checking which is the best set of those parameters to provide both: a tracing resolution much better than the reconstruction resolution; a reasonable computing time. Our approach to understand these tracing precision problem is twofold: Test A: verify that tracing an electron (with physics processes with material points switched off) produces accurate results when compared to an analytical calculation. Test B: check how accurately we get back to the starting point (tracing from left-to-right and then backward) as a function of the Geant4 parameters that basically control the tracing process.

3. Geant4 Tracking Test (D. Lunesu)3 Geometry used in the test: Intersection points The electron is traced from a starting point up to the end of the world box surrounding the pixel detector. It’s trajectory is then simply reversed and the intersection of the forward and backward trajectories with the first pixel plane is computed. We also compute the returning point using an analytical prediction. X Y Z Fig 1

4. Geant4 Tracking Test (D. Lunesu)4 Number of pixel planes: 21 Dimension of each plane: 1x100x100 cm 3 Distance between planes: 5 cm Fiducial volume surrounding the detector: 300x200x300 cm 3

5. Geant4 Tracking Test (D. Lunesu)5  Chord: max distance between the trajectory arc and the straight line connecting the two arc ending points.  OneStep: accuracy for the endpoint of ‘ordinary’ integration steps, which do not intersect a volume boundary.  Intersection: accuracy with which the intersection with a volume is calculated.  Chord Definition of the parameters used in test: test is made with several values of three G4 parameters that control the tracing: Fig 2

6. Geant4 Tracking Test (D. Lunesu)6 This test is designed to verify the agreement between the analytically computed trajectory and the Geant4 traced trajectory. Test particle : an electron (kinetic energy = 1GeV), with physics processes switched off, only transport in magnetic field. Uniform Magnetic Field : 1 T The electron is traced from left to right and from right to left to measure the amount of accumulation of rounding-off errors during the tracing process. The starting point is always in the same location for all events. The difference between the intersection of the backward trajectory and the corresponding analytical prediction at the first pixel plane is plotted for different values of  Chord and  Intersection (we will show later what happens under different conditions) Test A

7. Geant4 Tracking Test (D. Lunesu)7 The analytical track is calculated solving the differential equations: Solution:

8. Geant4 Tracking Test (D. Lunesu)8 va, vr ya yr Starting point x y Magnetic field ( 1T ) In this picture we define the quantities used in the following plots: va: the y coordinate computed analytically at the x of the first pixel plane in the forward direction vr: the y coordinate computed analytically at the x of the first pixel plane in the backward direction (always the same as va) ya: the y coordinated traced in the forward direction by GEANT4 (with a specified set of  Chord,  Intercept and  Step values. yr: the y coordinated traced in the backward direction by GEANT4, thus potentially accumulating rounding-off errors Fig 3

9. Geant4 Tracking Test (D. Lunesu)9 We observe, as expected, that the difference va-vr (computed analytically in our program) turns out exactly zero (plot 4a) The ya coordinate, always traced from the same starting point and same set of control parameters (see below) for all 244 events, not always turns out at the same value: we do NOT understand this behavior (plot 4c).  Chord = 0.099 mm  Intercept = 0.0047 mm  Step = 0.1 mm N.b: units of measure is mm in this and all subsequent plots a b c Starting point was y=120 mm for all 244 event in this plot Fig 4

10. Geant4 Tracking Test (D. Lunesu)10 Increasing the accuracy of the control parameters, the ya point turns out to be always the same (118.6 mm, plot 5c)  Chord = 0.01 mm  Intercept = 0.001 mm  Step = 0.1 mm a b c This behavior is totally unexpected, since the tracing is supposed to be a deterministic calculation: starting from the same point we should always reach the same intersection (at different values changing the control parameters, but keeping those fixed, the traced intersection should always be the same) Fig 5

11. Geant4 Tracking Test (D. Lunesu)11 We have generated a large sample (10 6 events) with a mesh of different values for the parameters that control the tracing in GEANT4. We have then measured the difference between intersection points of the trajectory with the first pixel plane for the forward track and for the backward track. Values we used are:  Chord = 0.01, 0.012, 0.013, …, 0.099 mm  Intercept = 0.0010, 0.0012, 0.0013, …, 0.0099 mm  Step = 0.1, 0.2, 0.3, …, 1. mm In the first plot of the next page, we plot the difference between an intersection point computed analytically and the same point reached after a complete tracing. This is done for the mesh of :  Chord vs  Intercept  Chord vs  Step  Intercept vs  Step 1 2 3... 72

12. Geant4 Tracking Test (D. Lunesu)12 YtYt YcYc  Intersection  Chord Ranges: 0.001 mm   Intersection  0.01 mm in step of 0.0001 mm 0.01 mm   Chord  0.1 mm in step of 0.001 mm The vertical axis of this scatter plot is the mean value of the following quantity: Y c is the intersection of the backward trajectory, with the first pixel plane, computed analytically Y t is this same intersection of the backward trajectory obtained after a complete tracing. YY  Y=Y c -Y t X of first pixel plane Fig 6

13. Geant4 Tracking Test (D. Lunesu)13  Intersection  Chord YY What we observe:  Y is almost independent from  Intersection, except for particular values of  Chord (see spikes in fig.4)  Y smoothly depends on  Chord.  Chord = 0.01 mm   Y ~ 0.2  m  Chord = 0.1 mm   Y ~ 3.0  m The value of  Chord needed to achieve a tracing precision below 1  m is 30  m (red dotted line in figure) We really do not understand the weird behavior of this correlation distribution. It seems that for particular values of  Chord the tracing precision degrades dramatically and a strong dependency from the  Intersection parameter ensues. Fig 7

14. Geant4 Tracking Test (D. Lunesu)14 In this picture one can notice the discrepancy between the forward and the backward trajectory Fig 8

15. Geant4 Tracking Test (D. Lunesu)15 YfYf YbYb  Intersection  Chord YY The vertical axis of this scatter plot is the mean value of the following quantity: Y f is the intersection of the forward trajectory with the first pixel plane Y b is the intersection of the backward trajectory computed after a complete forward-backward tracing.  Y=Y f -Y b X of first pixel plane Ranges: 0.001 mm   Intersection  0.01 mm in step of 0.0001 mm 0.01 mm   Chord  0.1 mm in step of 0.001 mm Fig 9

16. Geant4 Tracking Test (D. Lunesu)16  Step  Chord YY Ranges: 0.1 mm   Step  0.9 mm in step of 0.1 mm 0.01 mm   Chord  0.1 mm in step of 0.001 mm Fig 10 This plot shows the tracing accuracy (as defined in previous pages) as a function of both  Step and  Chord. As can be seen, there is no dependency on  Step, the only variation being due to  Chord. Again, though, there are specific values (marked in the figure by a red arrow) where particular values of  Chord give rise to an anomalously large value of the  Y quantity.

17. Geant4 Tracking Test (D. Lunesu)17  Step  Intercept YY Ranges: Fig 11 This plot shows the tracing accuracy (as defined in previous pages) as a function of both  Step and  Intercept. As expected, there is no dependency on both  Step or  Intercept: this is expected since this plot is just another view of the preceding ones. The big jump corresponds to the already observed anomalous behavior of the  Y quantity for specific values of one of the other parameters. 0.001 mm   Intersection  0.01 mm in step of 0.0001 mm 0.1 mm   Step  0.9 mm in step of 0.1 mm

18. Geant4 Tracking Test (D. Lunesu)18

19. Geant4 Tracking Test (D. Lunesu)19 CONCLUSIONS We do not understand why, with all parameters fixed to a particular set of values, the intersection of a track with the first pixel plane does not always give the same number (see fig. 4c, page 9). The tracing accuracy (defined as the average value of the difference between the intersection of a track computed in the forward direction with that computed in the backward direction) seems to depend only on the  Chord parameter and NOT on  Intersection or  Step, except for particular values of  Chord where a strong dependence on  Intersection ensues, see fig. 6, page 12. A tracing accuracy better than 1  m is obtained with  Chord <10  m regardless of  Intersection or  Step (fig. 7 page 13) Still missing in this study is the CPU time dependence for this set of parameters We are in close contact with the Geant4 development team to get help in resolving these issues