1. 1 Emittance Calculation Progress and Plans Chris Rogers MICE CM 24 September 2005

2. 2 About the Beard… It could have been worse…

3. 3 Overview Talk in detail about how we can do the emittance calculation –Sample bunch –Remove experimental error (PID & tracking) –Calculate Emittance Talk about other useful quantities –Scraping/Aperture –Decay Losses –Single Particle Emittance –Single Particle Amplitude –Holzer Particle Number

4. 4 Emittance(z) +/- error U i meas Sampled bunch PID V meas V true Emittance Calculation Roadmap Understood, tools exist Roughly understood Not really understood I’ll run through each box in this talk

5. 5 Beam Matching The cooling channel is designed to accept a certain distribution of particles –The beta function should be periodic over a cell of the magnetic field –The beta function should be a minimum in the liquid Hydrogen for optimal cooling –The longitudinal distribution should be realistic for the appropriate phase rotation system My standard approach is to –Do a reasonable job with the beamline (for good efficiency) –Then sample a Gaussian distribution from the available events for the final analysis Assign each muon some statistical weight w Matching Condition is usually (  = (333 mm, 0) in the upstream tracker solenoid (MICE Stage VI) –Note  =400 mm for p z = 240 MeV/c

6. 6 Sampling a bunch - stupid algorithm Stupid algorithm already exists but fails –Bin particles –Density,  bin = n bin /(bin area) –Apply statistical weight to all particles in bin W bin =  required  bin Fails because number events in each bin goes as –With 10 6 particles and 10 bins/dimension we have ~ 1 particle in each bin –Atrocious precision Should be possible to do better –Some algorithms planned but not implemented

7. 7 Momentum Amplitude Correlation At least three definitions of the amplitude exist –“Palmer” (FSII) –“Balbekov” (muc258) –“Ecalc9” (muc280) - note units are [mm] A prescription for generating the correlation exists –Generate transverse phase space in a gaussian as normal –Generate  E or  p z in a gaussian as normal –But add a term to make E or p z like Nothing more than a handwaving justification in the literature A prescription we can follow I guess But many questions remain

8. 8 More on P-A correlation Build grid in phase space all with p z 200 MeV/c e.g. –(x,p x ) = (0,5) (0,10) (0,15) … (5,5) (5,10) (5,15) … Fire it through MICE magnetic fields –No RF/LH 2 We introduce a momentum amplitude correlation!!! 15 pi 6 pi (FS2)

9. 9  meas We then calculate the covariance matrix using –Where u i are the measured phase space coords, and w is the statistical weight –I haven’t specified whether we use p x or x’=p x /p z type variables Recall emittance is related to the determinant of the 2N dimensional covariance matrix according to –Where the additional factor of is required if we use x’ type variables to normalise the emittance –And  is the matrix with elements  ij or

10. 10  true This gives us the measured covariance matrix –Includes errors due to mis-PID –Includes errors due to detector resolutions Correct for detector resolutions –Detector resolution introduces an offset in emittance –If we can characterise our detector resolutions well, we can understand and correct the offset Correct for mis-PID –Mis-PID also introduces an offset –If we can characterise our PID and beam well, we can in principle correct this offset

11. 11 Measurement Error The expression for  meas in terms of the error in the measurement of the phase space variable is given by For a bit more detail see MICE Note 90 (tracker note) This is similar to addition in quadrature, except that the error is not independent of the phase space coordinates –Error on emittance not only dependent on the resolution in a single phase space variable –Worry about whether an error in x introduces and error in p x –Worry about whether the error in x is greater at different p x –E.g. worry that the TOF resolution at the reference plane is highly dependent on the p z resolution of the tracker

12. 12 Uncorrected 4D Emittance (Ellis)

13. 13 Corrected 4D Emittance (Ellis)

14. 14 Calculating the Error Calculate the values of R, C using G4MICE –Verify G4MICE in stage I & II –Knowing the error is more important to the baseline analysis than the actual size of the error Requires some care –Once we are beyond MICE stage I & II it will be difficult to re- verify G4MICE –If the detector errors change we will be blind E.g. the spectrometer field drifts, a fibre dies, etc… So we should understand the errors in detail –So, for example, if the B-field drifts during the experiment we can spot it We should be actively checking sources of error –Check spectrometer field between runs, etc…

15. 15 PID Error introduced on covariance measurement by mis-PID is something like –bs is background identified as signal, sb is signal identified as background –This should be after the bunch sampling We should be able to estimate this offset –We can measure the distribution of incoming particles –We can calculate the probability of mis-identification (from e.g. Monte Carlo simulation) In the case that V ij bs, V ij sb ~ V ij true emittance is not changed –N meas =N true +N bs -N sb We should also worry about measurement of transmission –Perhaps this is more important for PID –We need to understand what analysis is required for scraping These ideas need to be verified by simulation

16. 16 Useful Quantities - Scraping There exists a closed surface in phase space beyond which particles strike the walls –Surface in 6D phase space We should be able to measure this surface –Transmission, radiation damage, ?dynamic aperture?, ?rf bucket? We should be able to measure the effects on the muon of striking the walls –Are all particles lost? This means that we must have sufficient acceptance in the detectors etc that the entire scraped surface makes it to the first absorber It would also be useful to distinguish between scraping losses and decay losses –Is this possible?

17. 17 Useful Quantities - Decay Losses We may also want to get at decay losses Expect ~ 20% or more loss in a FS2 style neutrino factory cooling channel due to decays But should be easily calculable

18. 18 Useful Quantities - SPE Single Particle Emittance  i –V is the matrix of covariances –U is particle position –O is the matrix of measured optical functions , , etc V=  n O –Can be calculated in G4MICE Analysis SPE is area of this ellipse Position of particle RMS contour of bunch

19. 19 Constancy of SPE Fire a 5  beam through MICE stage VI with only magnetic fields –No RF/liquid Hydrogen –Individual SPE’s change by ~10 %, ~ constant

20. 20 Cooling ito SPE Now add RF (electrostatic?) and LH 2 –Note decreases by ~10% … Cooling!! (  ~ /2n)

21. 21 Useful Quantities - SPA SPA single particle amplitude ~ Ecalc9f amplitude above –Calculate optical functions O c –SPE-like quantity independent of bunch measurement –Can be calculated in G4MICE Analysis –Note O c is not uniquely defined (depends on input beam) One powerful use of this method is to look at phase space without requiring any bunch –Good for simulation –Possibly use as an experimental technique? –Get much higher statistics in particular regions of phase space Get back to “bunch amplitude” ~ bunch emittance –Use

22. 22 Example use - nonlinear optics Build grid in phase space as above Fire it through MICE magnetic fields –Examine change in amplitude upstream vs downstream –No RF/LH 2 Show nice features –Dynamic aperture? –Emittance growth vs (x,p x,y,p y )? Initial A 2  A 2 /A 2 Initial x  A 2 )/A 2

23. 23 Useful Quantities - Holzer Emittance Calculate the maximum number of particles sitting in an arbritrary hyper-ellipsoid of a given volume –Holzer suggests using a minimising algorithm to find the hyper- ellipsoid of a particular volume that has the most particles in it –To first approximation, this will be similar to the hyper-ellipsoid given by U T V -1 U –Then this becomes the number of particles with SPE lower than some value ~ the volume of the hyper-ellipsoid

24. 24 Unanswered Questions How do we do the offline bunching? –I have some ideas –This is the next thing to tackle Analysis of longitudinal dynamics –We need to understand the TOF resolution ito 6D emittance measurement –How do we do the 6D emittance measurement? How good is the PID in terms of emittance? –Do we need the correction/will it really work? Does there exist a serious understanding of momentum- amplitude correlation? –We cannot really talk about this without understanding –We need a detailed analysis of the non-linear beam dynamics of the cooling channel More detail on the scraping/transmission analysis