Implementation of Covariance Matrix on ReconstructedParticle C. - - PowerPoint PPT Presentation

Implementation of Covariance Matrix on ReconstructedParticle

• C. Calancha

ILD Analysis & Software Meeting April 16, 2014

• C. Calancha (KEK)

Covariance Matrix April 16, 2014 1/11

Motivation

ReconstructedParticle.getCovMatrix is not implemented in current ILCSOFT release (return 0, ∀ p ∈ PandoraPFOs ) . This method provide covariance matrix of the reco. particle 4 vector {px,py,pz,E}. I was suggested to apply this cov. matrix to obtain dimuon mass error event-by-event basis. I have written Marlin processor adding new LCCollection to the event. This collection is a copy of PandoraPFOs but with filled cov. matrix. Code is available here: http://www-jlc.kek.jp/jlc/en/node/209

• C. Calancha (KEK)

Covariance Matrix April 16, 2014 2/11

Calculation

Σi = Σi(tanλ, φ, Ω), Σ′

i = Σ′ i(px, py, pz, E)

Covariance matrix on helix parameters, Σi, from associated track. Obtain jacobian (J) and perform: Σ′

i = JT Σi J

• C. Calancha (KEK)

Covariance Matrix April 16, 2014 3/11

Dimuon invariant mass error

Checked calculation with muons from H → µµ . The covariance matrix is used to obtain the dimuon invariant mass event-by-event.

• C. Calancha (KEK)

Covariance Matrix April 16, 2014 4/11

)

H

θ cos(

• 1
• 0.5

0.5 1 ]

2

)) [GeV/c µ , µ (M( σ 0.2 0.4 0.6 0.8 1

ILD Preliminary

Better precision at central region (tracks have more hits).

• C. Calancha (KEK)

Covariance Matrix April 16, 2014 5/11

H

φ

• 2
• 1

1 2 ]

2

)) [GeV/c µ , µ (M( σ 0.2 0.4 0.6 0.8 1

ILD Preliminary

No dependence on azimutal angle.

• C. Calancha (KEK)

Covariance Matrix April 16, 2014 6/11

Gaussian Fit in [-2,2]

)) µ , µ (M( σ ) - 125) / µ , µ (M(

• 20
• 10

10 20 200 400 600

ILD Preliminary

Gaussian fit [-2,2]: mean: -0.157 sigma: 1.0001

• C. Calancha (KEK)

Covariance Matrix April 16, 2014 7/11

Summary / Plan

Summary Current ILCSOFT does not provide covariance matrix on {px,py,pz,E} for the reco. particles. Developed Marlin processor calculating this matrix for charged particles. Plan Use cov. matrix in update of H → µµ analysis.

• C. Calancha (KEK)

Covariance Matrix April 16, 2014 8/11

BACKUP

BACK UP

• C. Calancha (KEK)

Covariance Matrix April 16, 2014 9/11

Relation between variables

Original base: A = {tan λ , Ω , φ , d0 , z0 } New base: B = {px , py , pz , E} px = pT cosφ py = pT sinφ pz = pT tan λ E2 = (a Bz Ω cos λ)2 + m2 = ( pT cos λ)2 + m2 pT = | κ Ω | κ = |a Bz| (constant) Momenta does not depend on d0 , z0 px = px (tan λ , Ω, φ) py = py (tan λ , Ω, φ) pz = pz (tan λ , Ω) Change of cov. matrix

1

Σ′

i = JT Σi J 2

Σi cov. matrix in A.

3

Σ′

i cov. matrix in B.

• C. Calancha (KEK)

Covariance Matrix April 16, 2014 10/11

Jacobian helix parameters to momenta space

After some derivative exercises ... J =                

∂Px ∂tanλ ∂Py ∂tanλ ∂Pz ∂tanλ ∂E ∂tanλ ∂Px ∂Ω ∂Py ∂Ω ∂Pz ∂Ω ∂E ∂Ω ∂Px ∂d0 ∂Py ∂d0 ∂Pz ∂d0 ∂E ∂d0 ∂Px ∂z0 ∂Py ∂z0 ∂Pz ∂z0 ∂E ∂z0 ∂Px ∂φ ∂Py ∂φ ∂Pz ∂φ ∂E ∂φ

                = −1 Ω                  −ΩPT − P2

z Ω

E tanλ

Px Py Pz

P2 E

Py Ω −Px Ω                 

→ Σ′

i = JT Σi J, covariance matrix in momenta space.

(Σ′

i = J Σi JT if you define jacobian as the transposed of quoted above)

• C. Calancha (KEK)

Covariance Matrix April 16, 2014 11/11