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

2. Overview Matching between beamline and MICE may be difficult Suggest reweighting algorithm to realign beam “offline” Apply continuous polynomial weighting Enables choice of beam moments at input to MICE => emittance, beta function, alignment, amplitude moment corr etc Discuss 1 dimension case Demonstrate extension to 2 dimensions (and more) Reweighting is necessary for ANY measurement of cooling Perhaps except at Step 4 (one absorber only) Amplitude analysis DOES NOT save us Analysis application for online analysis Histogramming (and graphing) of useful parameters GUI’d Cuts on useful parameters Reweighting using above algorithm to be implemented Online optimiser interfaces to many codes

3. Alignment Requirement Requirement on matching and alignment of beam As measured at the tracker reference plane Answer “how well matched should the beamline be to MICE” MICE note to be published soon Variable1% Cooling Requirement10% Cooling Requirement 2 mm6 mm 2 MeV/c6 MeV/c 2 MeV7 MeV 50 mm 2 200 mm 2 Corr(x,p x )0.040.1

4. Addendum - Solenoid/MICE alignment PRELIMINARY Fire a particle at z = -7000 mm with some px and x from beam axis Measure position at tracker reference plane What is the misalignment induced by traversing the solenoid fringe field? How well should the beamline be physically aligned to the tracker solenoid R [mm]Pt [MeV//c]

5. Reweighting The beamline cannot produce what we require Need amplitude momentum correlation for 6D cooling Will 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 Weight using a polynomial series

6. 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 using the simultaneous equation 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 2n th moment from input distribution

7. 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 input (Line) Parent pdf (Hist) Unweighted events Output (Line) Expected analytical Pdf (Hist) Weighted events Mean = 0.1 Mean = 0.0 Variance = 1 Variance = 0.9

8. Extension to Many Dimensions It is possible to extend the technique to many dimensions For position variables x i and N dimensions If we calculate input moments V i…i f and choose output moments V i…i g then a i…I can be calculated using the relation This is just a big(!) simultaneous equation which can be solved using a big~n N matrix inversion where N is the largest moment and n is the number of dimensions

9. Implemented Prototype ND Code Consider uncorrelated 2D distribution Introduce a correlation Technique works to machine precision For 2nd moments anyway Maximum order moment I can choose? Largest offset in moment I can introduce? How does it affect statistical error? This is gorgeous! Unweighted m=2 (choose means & covariances) m=4 (choose 1st through 4th moments)

10. Covariance Matrix Look! Magic! Means: (0.029, 226.07) Covariances: 1.07051 -0.0641778 -0.0641778 0.940308 Means: (1e-17, 226.000000) Covariances: 1.5000000 0.5000000 0.5000000 1.0000000

11. 11 Emittance at TRPs +/- error Measure x, y, px, py, E at TRPs Choose beam at upstream TRP Upstream PID Measure beam at downstream TRP Calculate true covariances Measure/calc t at TRPs Downstream PID TRP = tracker reference plane ? ? ? ? Analysis Roadmap

12. 12 Emittance at TRPs +/- error Measure x, y, px, py, E at TRPs Choose beam at upstream TRP Upstream PID Measure beam at downstream TRP Calculate true covariances Measure/calc t at TRPs Downstream PID TRP = tracker reference plane ? ? ? Analysis Roadmap

13. Conclusions A powerful reweighting algorithm Looks very encouraging Given a reasonable input distribution of muons, we can Choose input emittance to machine precision Choose input , , angular momentum, etc to machine precision Choose amplitude momentum correlation to machine precision What is a reasonable input distribution? How far can we push this algorithm? Fire a reweighted beam down the beamline? What are the effects on statistical error? Next step Full analysis of MICE step VI or IV Perhaps excluding PID? Using realistic beam Aim is to be ready to publish as soon as we have muons!

14. Online GUI Online Analysis GUI

15. FINISH END

16. 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=16 Output N=12

17. 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 But who needs 14 th moments anyway This is a very successful technique In principle this technique can be extended to 6D phase space Matrix becomes larger But inverting a matrix is easy?