1. Particle by Particle Emittance Measurement to High Precision Chris Rogers Imperial College/RAL 17th March 2005

2. Overview Detail the resolution of the MICE tracker in measuring phase space variables See this creates a bias in the emittance measurement Propose a correction routine Performance plots by Malcolm Ellis

3. Bunch Emittance Define the bunch emittance in some 2N dimensional space by Where V is the covariance matrix with elements Here U is the position vector in some space The MICE baseline is to take U to be the phase space vector e.g. (t,E,x,p x,y,p y ) This is analogous to the volume of a hyper-ellipsoid in 2N dimensions 2N th root gives it area-like dimensions Use of mechanical phase space makes it self-normalising Emittance resolution is determined by resolution in u i

4. Resolution of Tracker - x, y  Transverse position resolution  Difference between x and y caused by threefold rotational symmetry in tracker design  At equilibrium emittance ~ smallest emittance we wish to measure rms ~ 2.5%

5. Resolution of Tracker - px, py Resolution in transverse momenta RMS ~ 10%

6. Emittance Bias - approx We want to relate the resolution in our phase space variables to emittance First approximation -> addition in quadrature Hence we can recover the true emittance from the measured emittance Except for statistical fluctuations from the error pdf and the true pdf (Central Limit Theorem) In order to be sure of factoring this offset out we require: Say  2 erro r <~15% of  2 meas to give a bias in  2 meas <1% Our transverse phase space variables have achieved this

7. Emittance Bias - detail In the detail it turns out the error pdf is not independent of the true pdf Originates from Kalman? From dead fibres? Recall the covariance matrix has elements Writing u i meas =u i true +du i error and substituting into the equation for covariance we find

8. Emittance Bias - transverse We get an offset in emittance in our Monte Carlo Different in upstream and downstream trackers Can be negative or positive Different for different emittances upstream downstream

9. Emittance Bias - detail We can write this offset in matrix form… Where R has elements This is the term that comes from a correlation between the true pdf and the errors Then the emittance can be calculated from |V true |

10. Corrected Emittance By using the formula for the bias and taking the resolutions from our monte carlo we can correct the offset upstreamdownstream

11. Resolution of Tracker - E Energy resolution is just below 14% Again, taken at equilibrium emittance We do not have good figures for our TOF resolution yet Subsequent slides will show emittance as a function of TOF resolution

12. Emittance Bias - Longitudinal This is the uncorrected emittance bias as a function of TOF resolution In FSII the bunch has  (E)~ 25 MeV  (t) ~ 500 ps We measure the resolution with this bunch in mind

13. Corrected Emittance We apply the same correction technique This is the corrected emittance resolution as a function of TOF resolution

14. Summary Simulation at equillibrium emittance with 510 ps RMS in time and 25 MeV RMS in Energy RMS of the Monte Carlo truth distribution RMS of the (Reconstructed – Monte Carlo truth) distribution Ratio of the RMS of the (Reconstructed – truth) distribution to the RMS of the Monte Carlo truth distribution RMS of the (Reconstructed – Monte Carlo truth) distribution if the tracker volume contains He gas X21.00.5432.58 %0.542 Y21.00.4422.10 %0.440 PX17.92.0511.5 %2.05 PY17.91.528.52 %1.52 E25.23.4913.8 %3.48  We have seen the following tracker resolutions:  We understand how to relate these resolutions to emittance

16. Cartoon MICE Stage III Coil -> 4T constant B z Tracker