1. MICE input beam and weighting Dr Chris Rogers Analysis PC 05/09/2007

2. Overview Recap - MICE input beam alignment & matching X, y, px, py misalignment effect on cooling Energy “misalignment” effect on cooling Beta, alpha mismatch effect on cooling Dispersion effect on cooling cooling Third (& higher) moment effect on cooling A possible reweighting algorithm to realign beam “offline” Apply continuous polynomial weighting Discuss only 1D case but extensible to 6D etc Choose desired output moments => output emittance, alignment, amplitude moment corr etc

3. Alignment Sensitivity

4. Energy Alignment sensitivity

5. Linear Mismatch Sensitivity

6. Dispersion Sensitivity

7. Amplitude-Momentum Corr

8. Energy Dependent Beta

9. Reweighting What if we don’t get the desired beam May need to reweight input beam This is true for bunch emittance and particle amplitude analyses Reweighting in 6D is difficult No real way to measure particle density in a region Binning algorithms break down as phase space density is too sparse in high-dimensional spaces FT/Voronoi type algorithms seem to become analytically challenging in > 3 dimensions If I can’t measure density I can’t calculate weight needed to get a particular pdf Propose a reweighting algorithm based around beam moments Beam optics can be expressed purely in terms of moments of the beam No need to discuss actual pdfs at all Weight using a polynomial series and assess the quality of the weighting by looking at the moments before and after

10. Reweighting Principle Say we have some (1D) input distribution f(x) with known raw moments like f, f etc Say we have some desired output distribution g(x) with known raw moments like g, g etc Apply some weighting w(x) to each event so that Then the a i can be found in terms of input and output moments analytically Say we calculate coefficients up to a N Then N is the largest moment that we can choose in the target distribution Then we need to invert an NxN matrix And we need to calculate a 2Nth moment from input distribution Some maths details which I don’t reproduce here

11. Reweighting effects For 10,000 events, N=12 Input gaussian with: Variance 1 Mean 0.1 Output gaussian with moments: MomentTargetActual 100 20.9 300 42.43 610.935 868.89168.8905 10558.01 115524.3 121041.97948.22 output input (Line) Parent pdf (Hist) Unweighted events (Line) Expected analytical Pdf (Hist) Weighted events

12. Technique goes awry for large N Largest coefficient calculated is a N As I ramp up N the technique breaks down Numerical errors creeping in Can compare output calculated moment with target moment to find when the technique breaks down Output N=8 Output N=16

13. Failure vs N Consider output moment/target moment “Relative error” See a clear transition at N=12 What is the cause of the failure? Calculation of moments? May be a better way Inversion of matrix? I am using CLHEP for linear algebra Better linear algebra libraries exist This is still a feasible algorithm In principle this technique can be extended to 6D phase space Matrix becomes larger 6x6 for 1st moments ~24x24 for 2nd moments ~200x200 for 3rd moments But inverting a matrix is easy?

14. Conclusions Some study of alignment and matching sensitivity of MICE Perhaps conflicts with earlier studies Needs to be resolved Perhaps needs another look with higher statistics A proposal for a reweighting algorithm Looks encouraging in 1D Some computational error for reweighting the tails of the distribution See how it extends up to higher dimensional phase spaces