The North and South muon arms must be aligned relative
to each other if vector mesons which produce one muon which
goes into the North arm and one muon which goes into the
South arm are to be reconstructed. The alignment requirement
is determined by the opening angle resolution that is required
to get a good mass resolution of these particles. Alternatively,
we can make sure that the error in the angular measurement
due to misalignments is much less than the error caused by
multiple scattering in the absorber materials so that the
opening angle resolution will be determined solely by the
absorber material present. For the upsilon vector meson
mass resolution, the error in the opening angle measurement
caused by multiple scattering in the absorber material
(which is on the order of 15 mRad) is comparable to the
opening angle resolution that would be required if it were
to not contribute significantly to the mass resolution.
(If the error in the mass resolution caused by an opening
angle error is to be less than 1%, then the angular
resolution that is required for an opening angle of 110
is 14 mRad.) The multiple scattering caused by just the
magnet steel is approximately 9 mRad, so the multiple
scattering limit if the copper nosecone were not present would
be similar.
Because the multiple scattering limit is as small or smaller than
the limit that would cause the opening angle error to
contribute significantly to the mass resolution
we will determine the alignment requirements
by requiring that the opening angle measurement error caused
by misalignments be much less than the error caused by
multiple scattering.
Since the arms will be independently required to be aligned to the vertex to get good momentum resolutions in the two arms the z alignment of the two arms relative to each other should be taken care of. If in addition the two arms are required to be axially aligned to the beam line, and azimuthally aligned to some common x-y plane target, then the opening angle measurment should be assured. The theta and phi resolutions that are obtained for upsilon decay muons with all of the absorber materials present inside the North and South muon arms are approximately 0.4-0.6 mRad inside the tracking volume, and 3-9 mRad at the vertex as shown in Figure 7. To retain this angular resolution at the vertex, the displacement of the North arm with respect to the beam axis should be much less than (1.8 meters)*tan(3 mRad)=5.4 mm, where 1.8 meters is the z distance of station 1 from the interaction point. Similarly, the phi alignment with respect to a common (to both arms) alignment point should be much less than 3.0 mRad for the upsilon criterion. This corresponds to a displacement at the outer radius of station 1 chambers of (1.58 meters)*tan(3 mRad)=4.8 mm. We therefore require that the two muon tracking arms be aligned relative to a common survey point to 5.4/4=1.4 mm in the r direction, and to 4.8/4=1.2mm in the phi direction at the outer radius of station 1 (or to within 0.8 mRad=2.8 arcmin). The z alignment that is required for the individual arms will be more than adequate to provide the arm-to-arm alignment required.
Figure 7: The theta and phi resolutions obtained by
tracking muons from upsilon vector mesons through the
two muon tracking arms to the vertex. A vertex resolution
of 1 mm was used, and the copper nosecone, magnet steel,
and copper piston plug absorbers were in place.