MIPP TOF Studies

I've been looking at Ed's simplified reconstruction.  In particular, I wanted to add the effect of the CKOV and RICH gas, to get a more realistic idea of the ultimate momentum resolution.  (I know - to do this right I should use the Monte Carlo, but I can include these two big sources easily in the spreadsheet model and get a sense of what is important.)

Ed's Study

Ed's study is linked off the TOF page.  In short he considers the Rosie spectrometer in three pieces, two straight line track fits to the chamber hits upstream and downstream of Rosie, and the transverse momentum kick in Rosie, ptR.  The momentum measurement consists of measuring the difference in track angle between the two track segments, Delta x' , and comparing to ptR:

pz = ptR / Delta x'

The issue of momentum resolution is reduced to the error in determining the slopes of the track segments.

Tracking Through Rosie

Ed tracks trajectories through Rosie using the scheme presented in his tracking note.  For particles with initial pt = 0:

Plot of trajectories through Rosie.

Corrections to Ed's Study

In the process of understanding Ed's study, I think I've found two errors.

First, his expressions on page 2 for the errors on the track fit parameters should be normalized by the determinant of the error matrix.  This increases the momentum error estimate.  For the errors on the track parameters Ed writes

Ed's expression for the errors

The correct expression (from Bevington, for example) is

Sigmas from Bevington

where the normalization factor is the determinant of the error matrix.  Inverting and expanding these expressions one finds Ed's expressions as the first terms.

Second, his formula for the error on pz, at the top of page 3, gets quite large for pt = 0, as he noted, but this is incorrect.  (His form gives a large error when either one of the slopes is near zero, whereas only the slope difference matters - rotational invariance and all that.) The correct expression is to take the fractional momentum error equal to the fractional error in the track angle difference:

pz variance

Both of these corrections are included in the results below, and in this revised Excel workbook.

Impact of MCS in the CKOV and RICH

As a first order study of the impact of MCS in the CKOV and RICH, I estimate the impact of each on the track angle measurement on its side of Rosie.  To do this, I've  taken the mean MCS angle in the detector (gas only, no windows or mirrors) and projected to the spot size at the next chamber.  I take this MCS spot size in quadrature with the chamber spot size as the error on the transverse position measurement at that chamber.  I continue to grow the spot size to any following chambers.  I don't propagate the MCS growth through Rosie, since upstream and downstream are separate track angle measurements.

In Ed's treatment of MCS (his "MCS" worksheet) he generates a random ray from the MCS distribution and tracks it explicitly through the magnets.  I implicitly assume the MCS is small, so I only carry along the width of the distribution, which I assume is centered on the unscattered reference trajectory.  Ed computes a more accurate trajectory for the selected ray, but gets no distribution information.  I use only the central trajectory but get to compute the spot sizes.

The effect of the TOF will be included as an additional error term in the track angle difference, added in quadrature with the slope fit errors.

Using what I think is the correct formula, I observe the following:

  1. Using the nominal chamber resolutions and no CKOV or RICH:

    1. The fractional momentum resolution is independent of  pt. - Good.
    2. Momentum error scaling = 2.839 x 10-4 (GeV/c)-1 independent of momentum, or sigma pz / pz  = 2.84% at pz = 100 GeV/c.
    3. The resolution comes from 138 µrad track angle error upstream of Rosie, and 71 µrad downstream at pz = 100 GeV/c.  The larger intrinsic resolution of the Iowa chambers is more than offset by the longer lever arm to DC6.

  2. Adding a 1 m Long CKOV

    3. There is no loss of resolution.  I modeled the CKOV as C4F10, X0 = 35.7 g/cm2, rho = 10.1 mg/cm3, x/X0 = 0.0284.  This results in 19.8 µrad at pz = 100 GeV/c.  Using 1 m thickness is a gross overestimate for the gas, since most of the tracks pass through the center where the window angles in behind the mirror planes, leaving more like 15 cm thickness.

  3. Adding a 10 m Long RICH

    4. I modeled the RICH as CO2, X0 = 36.2 g/cm2, rho = 2.17mg/cm3, x/X0 = 0.0600.

    1. Momentum error scaling is momentum dependent, smallest at 100 GeV/c, but the momentum resolution, sigma pz / pz, is roughly independent of momentum and less than 3% down to 50 GeV/c.
    2. This comes about because the downstream track angle error increases to 77 µrad at pz = 100 GeV/c.

  4. Adding a 5 cm Long TOF

    5. To compute the error on the momentum measurement, I add the TOF MCS in quadrature with the track angle difference error.  This ignores some geometry (the longitudinal offset between the TOF and the center of Rosie) which I think is negligible, since the reconstruction only depends on the angle difference, not the intercept coordinate of the two tracks.

    A 5 cm TOF adds 43 µrad error to the angle difference measurement. (X0 = 43.7 g/cm2, rho = 1.03 mg/cm3, x/X0 = 0.118.)  For results as a function of momentum see the table below.

  5. Using a 50 µm Resolution for DC6

    6. Using 50 µm resolution for DC6 has a very small effect, due to the detector gases.  See the table below.

  6. As a Function of Momentum

sigma pz / pz
870 µm DC6 Resolution 50 µm DC6 Resolution
pz No TOF
 5 cm TOF No TOF 5 cm TOF
10 5.89% 9.81% 5.77% 9.74%
20 3.89% 5.52% 3.69% 5.38%
30 3.35% 4.25% 3.11% 4.07%
40 3.14% 3.70% 2.88% 3.48%
50 3.03% 3.42% 2.76% 3.18%
60 2.98% 3.25% 2.70% 3.00%
70 2.94% 3.15% 2.66% 2.89%
80 2.92% 3.08% 2.63% 2.81%
90 2.90% 3.03% 2.62% 2.76%
100 2.89% 2.99% 2.60% 2.72%

Fractional momentum error vs. momentum.

Fractional momentum error / momentum vs. momentum.

These values are consistent with both Tim's MC study and Carl's analytic study.

So, what momentum resolution do we really need?  This is a physics question; those with physics interest need to weigh in!

PDB, 3/26/03