1 Statistics David Forrest University of Glasgow May 5 th 2009

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 How We Mean To Proceed W e assume that we will discover a formula that takes the form Sigma=K*(1/sqrt(N)) where K is some constant or parameter to be determined. How do we determine K? 1) First Principles: do full error propagation of cov matrices → difficult calculation 2) Run a large number of G4MICE simulations, using the Grid, to find the standard deviation for every element in the covariance matrix → Toy Monte Carlo to determine error on emittance 3) Empirical approach: large number of simulations to plot   versus 1/sqrt(N), identifying K (this work)

4 What I’m Doing 3 absorbers (Step VI), G4MICE, 4D Transverse Emittance I plot 4D Transverse Emittance vs Z for some number of events N, for beam with input emittance . I calculate the fractional change in emittance . I repeat ~500 times and plot distribution of all  for each beam. Carried out about 15,000 simulations on Grid (8 beams x 1700 simulations/beam plus repeats)

5 8pi – N=1000 events 

6 Checks - X, X’ beforeafter 2.5pi 0.2pi

7 Checks - X, X’ beforeafter 10.0pi 8.0pi

8 Checks – beta function Expected beta in absorbers ~420mm, solenoid 330 mm after matching 0.2  4.0  2.5  10.0 

9 Results Events  rmsSims 0.2   

10 Results-2 Events  rmsSims 3.0   

11 Results-3 Events  rmsSims 8.0  

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20 K values BeamK KK C CC

21   =K/sqrt(N)

22 Sans pencil beam

23 Physical Meaning (J Cobb) There is a physical meaning to this K value By usual error formula, assuming no correlations: So without correlations, we have this factor, normally >1, eg if f=-0.08, we get a factor of 1.29

24 Physical Meaning However, there are correlations between input emittance and output emittance, so we include a correlation factor, k corr. The  sim I measure includes this also.

25 Correlation factor Preliminary

26 How many muons do we need? We want to measure to an error of 0.1% Beam (pi mrad)No correlations (10 6 events) With correlations (10 5 events)

27 Conclusions We need of order 10 5 muons to achieve 0.1% error on fractional change in emittance Simulations in place for doing a toy Monte Carlo study, to propagate errors from elements of covariance matrix