Shortcomings of Muon Track-fitting in Present INO-ICAL Code

7/24/2019 Shortcomings of Muon Track-fitting in Present INO-ICAL Code

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Shortcomings of Muon Track-fittingin Present INO-ICAL Code

Kolahal Bhattacharya


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Outline

The state vector

Kalman filter algorithm

Propagator matrix in present ICAL code

Speculations done in past

How to calculate the F matrix elements


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The State Vector

Charged particles follow

The a!ove ma" !e written as

These two are constrained !" Standard algorithm #K$ method% &uon Swimmer

m(d2x/dt2)=c2q(v x B)

x ' '=c

P

ds

dz[x ' y ' B

x(1+x '2)B

y+y ' B

z]

y ' '=cP

ds

dz[(1+y '2)Bxx ' y ' B y+x ' B z]

ds

2

=dx2

+dy2

+dz2

x=(x , y , x ' , y ' ,q

P)

T


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Kalman ilter

State '(n

&easurement e(n

Assume ) w*

+ , ) *

+ , -

.efine /*,)w

*w

*T+0 1

*,)

*

*T+0 Ci

*,cov2xi

*3x-

*4

.enote xi*, estimated state at *thdetector plane

from the measurement done up to ithdetector plane i)* ,+ prediction 5extrapolation60 i,* 5filter6 and

i+* ,+ smoothing

xk= fk

1

xk

1+

wk

1=

Fk

1xk

1+

wk

1

mk= hkxk+ k= Hkxk+ k


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ormalism

extrapolation of state vector

'xtrapolation of ''C

Filtered state follows from minimi7ation of

The is constructed as !elow

It must !e minimi7ed w8r8t8

xkF

k1x

k1

Ckk 1

= Fk 1Ck 1k 1

Fk 1T

+Qk 1

xkk

2

2

[h xkk 1

+ Hkxkk xk

k 1 mk]

TVk

1[ hxk

k 1+ Hkxk

kxk

k 1 mk]

+ xkk xk

k 1

TCk

k 1

1xk

k xk

k1

xkk


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Solution

Filtered state

Filtered estimation error covariance

To use in the next iteration

xkk= xk

k 1+ Ck

kHk

TVk

1[mk h xk

k1 ]

Ckk=

Ckk

1

Ckk

1HkT

Vk+ HkCkk

1HkT

1HkCkk

1

Ck+ 1k

= FkCkk

FkT+ Qk


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Summar!

An apriori information of 5 6 for 5*396thla"er

Initiall"% we get it from a trac*3finder

:e extrapolate 5 6 to 5 6 using ph"sicalarguments contained in F8

is updated to

is used to get the filtered estimate of the state

xk1,Ck

1xk

1, Ck

1

xk1,Ck

1 xk ,Ck

xkk=f

1Ck

k [f

2mk+f3xk

k1]

Ckk1=

Ck

Ckk

Ckk


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The MINOS matri"

F= [1 0 z 0 0.5By z

2

0 1 0 z 0.5Bx z2

0 0 1 0 By z0 0 0 1 Bx z

0 0 0 0 1+]


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ICAL matri"


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Pro#lems $ith ICAL Track itting

The definition of F matrix has !een ta*en unalteredfrom &I;oined properl"8


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An Old Presentation

?& pointed this issues out in an earlier tal*

He too% got the @second hump@ in path length method


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Path Length Method


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%o$ to make amends

:e must deduce the elements of F matrix that will!e proper for ICAL8 The code should !e transparentto neutrinos from all 7enith angles8

This needs us to solve the .' and get the F matrix Seen so far #K$ and another anal"tical iterative

method8 ;ot sure which one to use or whether thatwould !e compati!le with the other parts of the code

If ever"thing else remains the same% onl" thischange ma" not improve path length methoddrasticall"8


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Ackno$ledgements & 'eferences

I am ver" grateful to &eghna and ;itali di for theirsupport to do this >o!8

I used the following references

5a6 .ata Anal"sis Techni(ues in High 'nerg"Ph"sics !" oc*% ;ot7% ?rote and #egler

5!6 Bohn &arshall Thesis for &I;

Shortcomings of Muon Track-fitting in Present INO-ICAL Code

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