TrackerAlignment

Alignment Definition

This section explains how the tracker alignment is defined.

• There are four places that alignment corrections are applied
  • A transform between the whole tracker and the nominal Mu2e detector coordinate system
  • A transform between each plane and the aligned tracker
  • A transform between each panel and the aligned plane
  • A correction to the end positions of individual straws and wires within a panel
• transforms
  • apply rotation before displacement
  • a rotation is defined as right-handed about the local x, then y, then z axes (rad)
  • add an offset in position (mm)

Tracker Geometry Build

Here is how the geometry is conceptually built up, including the alignment

1. Straws are placed within the panel
   • this uses the local panel coordinate system, defined as UVW, a right-handed coordinate system. These correspond to the XYZ coordinates of the Duke panel X-ray mapper, defined in Mu2e document 5703.
   • The straw direction points along the +U axis, which points from the High Voltage (HV) side of the DRAC electronics board to the Calibration (cal) side.
   • straw 0 is the longest, innermost straw, straw 95 is the shortest, outermost straw.
   • V points radially outwards, from the center of straw 0 to the center of straw 94.
   • W is perpendicular to the panel, and to U and V, pointing from the back of the base plate to the cover. This is roughly from the center of straw 1 to the center of straw 0
   • panel U=0 is defined by the nominal symmetry axis of the panel
   • pane V=0 is defined to be midway between innermost and outermost straw centers, between straws 47 and 48.
   • panel W=0 is defined to be midway between the straw layers, which is also midway between the innermost and outermost straw centers
2. Apply the straw alignment. Note that straws are not rigid bodies due to gravitational sag, electrostatic displacements and construction-induced distortions, so rigid-body transforms are not applicable. Instead, Straw alignment is defined in terms of displacements in V and W at either end (HV and Calibration) of the straw, where the straw intersects the manifold (radius=700 mm) and is held rigidly in place. The mylar envelope (straw) and wire have separate corrections. To first order, the wire may be approximated as a line between these points, as the gravitational sagitta expected for nominal tension is < 50 microns. The precise geometric description of the entire straw requires dynamic information such as the tension, High Voltage.
3. Apply the panel alignment of rotations and displacements. These are rigid body rotation and translations WRT the local panel UVW coordinate system.
4. Place the panel in the nominal plane. Starting with each panel oriented so UVW is aligned with Mu2e XYZ, this is achieved by rotating every other panel around the Y axis by 180 degrees, then rotating the nominal panel center to its nominal phi position around the z axis, and finally adding its nominal displacement in Z and radially outwards. The exact positions are shown in Mu2e document 888 Figure 9. By convention, the nominal plane is defined as plane 0, described in Mu2e document 888. The nominal plane coordinate system (XYZ) is centered on the plane, with directions as defined by the Mu2e detector coordinate system.
5. Apply the plane alignment of rotations and displacements, with respect to the nominal plane coordinate system.
6. Place the plane within the nominal tracker. This involves moving it along the z axis, and rotating half the planes around the Y axis by 180 degrees. The pattern of plane rotations is defined in Mu2e document 888. It amounts to building a station by rotating one plane around Y by 180 degrees, then rotating every-other station around Y by 180 degrees.
7. Place the tracker within the Mu2e detector coordinate system by applying the tracker alignment. The detector coordinate system is defined as the nominal (perfect) tracker coordinate system.

Alignment Distortions

This section explains how distortions to the alignment are applied in simulation

To perfect and quantify the performance of our data-driven alignment procedure, we need to produce simulated datasets that model the detector in a known, distorted state. We can then run the alignment procedure on those and test the output to see how well the correct (nominal) alignment was recovered, how many iterations it took, etc.

We characterize several kinds of distortions. Some are random, based on estimates of the construction accuracy and the installation/optical survey precision. Others are systematic, representing collection distortions that are known to be difficult to extract using data, and can have direct impact on our physics results. These so-called 'weak modes' were initially introduced under the name of 'global distortions' in reference to the alignment of the BaBar silicon vertex detector, as described in the paper Local Alignment of the BABAR Silicon Vertex Tracking Detector The term 'weak modes' was coined during the LHC era, as described in the paper Alignment of the CMS tracker with LHC and cosmic ray data.

In Mu2e, we use a script to generate initial misalignments by combining an arbitrary set of random misalignments and systematic biases on particular weak modes using this script, such as:

• ZSqueeze: this is a systematic distortion in the Z position of planes (or panels) as a function of Z: ie dZpos/dZ = constant
• RSqueeze: this is a systematic distortion in the R position of panels (or straws) as a function of their radius R: ie dRpos/dR = constant
• (X,Y)skew: this is a systematic distortion in the X or Y position of planes or panels as a function of Z: ie a parallelograming of the tracker, dX/dZ (dY/dZ) = constant
• Twist: this is a systematic distortion in the azimuth of a plane or panel as a function of Z; ie dphi/dZ = constant

Data are always generated in G4 using a perfectly aligned geometry. We can then reconstruct using a distorted geometry, and test to see how well the alignment procedure recovers the nominal geometry. This method is (nearly) equivalent to running G4 with distorted geometry and reconstructing assuming nominal, but allows testing many kinds of misalignment using a single input dataset.