MICE input beam weighting

Dr Chris RogersAnalysis PC05/09/2007

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

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

Variable 1% Cooling Requirement 10% Cooling Requirement

<x> 2 mm 6 mm

<px> 2 MeV/c 6 MeV/c

<E> 2 MeV 7 MeV

<x2> 50 mm2 200 mm2

Corr(x,px) 0.04 0.1

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]

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

Reweighting Principle Say we have some (1D) input distribution f(x) with known raw moments like <x>f, <x2>f etc Say we have some desired output distribution g(x) with known raw moments like <x>g, <x2>g etc Apply some weighting w(x) to each event

so that

Then the ai can be found using the simultaneous equation

Say we calculate coefficients up to an

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

)(...)1()( 33

221 xfxaxaxaxg

...)1()( 33

221 xaxaxaxw

gj

fj

if

jig

jf

ii xxxxxa )(

Reweighting effects For 10,000 events, N=12

Input gaussian with: Variance 1 Mean 0.1

Output gaussian with moments:

Moment Target Actual

1 0 0

2 0.9 0.9

3 0 0

4 2.43 2.43

6 10.935 10.935

8 68.891 68.8905

10 558.01 558.01

11 5524.3 5524.3

12 1041.97 948.22

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

Extension to Many Dimensions

It is possible to extend the technique to many dimensions For position variables xi and N dimensions

If we calculate input moments Vi…if and choose output moments Vi…i

g then ai…I can be calculated using the relation

This is just a big(!) simultaneous equation which can be solved using a big~nN matrix inversion where N is the largest moment and n is the number of dimensions

)(...)1()(2

1 2

121

1

11xfxxaxaxg i

N

i

N

iiii

N

iii

gjj

fjj

N

ii

fjjii

gjj

fiiii mm

k

mkmkkVVVVVa ......

.................. 11

1

11111)(

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)

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

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

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

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!

Online GUI

Online Analysis GUI

FINISH

END

Technique goes awry for large N

Largest coefficient calculated is aN

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

OutputN=12

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 14th 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?