Statistics Toy Monte Carlo

1. StatisticsToy Monte Carlo David Forrest University of Glasgow

2. The Problem We calculate 4D emittance from the fourth root of a determinant of a matrix of covariances...We want to measure fractional change in emittance with 0.1% error. The problem is compounded because our data is highly correlated between two trackers.

3. G4MICE Study-1 • Ran ~500 simulations for N events for 8 beams, where N=1000,2000,10000 using G4MICE (tens of thousands of simulations on Grid) • Got distributions of fractional change in emittance • Acquired s for differentbeams at differentnumbers of events 8pi

4. G4MICE Study-2 • Plot s against 1/sqrt(N)-> proportionality • The gradient of thisline, K, can tell ushow many events werequire to make a precisemeasurement • Full details in MICE CM24 presentation • Conclusion: Require ~105 muons

5. Toy Monte Carlo • Wish to confirm results using a toy monte carlo and understand origin of results • Have modelled energy loss and multiple scattering in absorbers (incl windows) • Do not have acceleration, proper transport of beam, etc which you get with G4MICE • Do have, so far, 3000 muons/sec (generation-selection-measurement-propagation through experiment-measurement)

6. Generation Calculating random vectors (x, px, y, py) describing particles, within envelope defined by some covariance matrix Selection Ensure each of the parameters in our vector are gaussian. (the collection of many of our vectors becomes a multivariate gaussian) Model Experiment Only model the absorbers, in series. Essentially beam starts before the first absorber and ends after the last absorber, with no cavities in between. Calculate Emittance Before and after - > fractional change in emittance. Plot identical distributions to G4MICE study, get s for each beam and each number of events, get K for each beam, compare with G4MICE

7. Generate • Have rms for x, Px, y, Py, from an ideal covariance matrix for each beam • Using CLHEP randomisation package • Uniformly sample (selection, described in the next slide, retains x, px, y, py only when they fit a gaussian) • Generate many vectors of (x, px, y, py)

8. Selection The probability distribution function (which is a gaussian) is as follows: X is our vector (x,px,y,py), V is our covariance matrix, m is (0,0,0,0) f is between 0..1. We keep X when a random number is below f. This selects x,px,y,py with proper gaussian distributed values. We keep N such X vectors.

9. Model Experiment • Absorbers (not to scale at all. Correct scale in toy mc ) At each arrow I calculate energy loss and then multiple scattering as additive adjustments to px and py only. I do not ever drift x, y.

10. Model Experiment 2 Equation for energy loss: Equation for multiple scattering: Q0 here is the s for a distribution from which we sample Q, in a gaussian manner, and scatter with this Q

11. Calculate Emittance • You could get a 4x4 covariance matrix from every X=(x,px,y,py), ie for every muon. • But I want emittance for the full sample of say 10,000 such muons. I take the mean value* for <x2>,<xpx>,<xy>,<xpy>…each of those, get a 4x4 covariance matrix and calculate e * Should this be the mean or the variance?

12. Issues • Not comparing like for like (beam from a matcher card I used vs beam from matrix, not measuring in trackers but absorbers) – could fix these problems directly by doing a fresh run of 3x500 sims in G4MICE on Grid (~1 day) with the same matrix. • Have modelled Pz as 200 MeV/c for all particles, whereas Px, Py are randomly generated for each particle. I haven’t dropped Pz after each absorber and perhaps I really should. • Is covariance matrix valid for start of absorber ? • Other assumptions which have been described in the previous slides

13. Results so far…. • G4MICE K=0.293 • ToyMC K=0.132536 • So far do not agree, so need to keep working on this …