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 Σ i = Σ i ( tan λ, φ, Ω) , Σ ′ i = Σ ′ i ( px , py , pz , E ) Covariance matrix on helix parameters, Σ i , from associated track. i = J T Σ i J Obtain jacobian ( J ) and perform: Σ ′ 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

ILD Preliminary 1 ] 2 )) [GeV/c 0.8 µ , 0.6 µ (M( σ 0.4 0.2 0 -1 -0.5 0 0.5 1 θ cos( ) H Better precision at central region (tracks have more hits). C. Calancha (KEK) Covariance Matrix April 16, 2014 5/11

ILD Preliminary 1 ] 2 )) [GeV/c 0.8 µ , 0.6 µ (M( σ 0.4 0.2 0 -2 -1 0 1 2 φ H No dependence on azimutal angle. C. Calancha (KEK) Covariance Matrix April 16, 2014 6/11

Gaussian Fit in [-2,2] ILD Preliminary 600 Gaussian fit [-2,2]: 400 mean: -0.157 sigma: 1.0001 200 0 -20 -10 0 10 20 µ µ σ µ µ (M( , ) - 125) / (M( , )) 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 , d 0 , z 0 } New base: B = { p x , p y , p z , E } p x = p T cos φ p y = p T sin φ p T = | κ p z = p T tan λ Ω | B z E 2 = ( a Ω cos λ ) 2 + m 2 κ = | a B z | ( constant ) p T cos λ ) 2 + m 2 = ( Change of cov. matrix Momenta does not depend on d 0 , z 0 i = J T Σ i J 1 Σ ′ p x = p x ( tan λ , Ω , φ ) p y = p y ( tan λ , Ω , φ ) 2 Σ i cov. matrix in A . p z = p z ( tan λ , Ω) Σ ′ i cov. matrix in B . 3 C. Calancha (KEK) Covariance Matrix April 16, 2014 10/11

Jacobian helix parameters to momenta space After some derivative exercises ...  − P 2  z Ω  ∂ P y  ∂ P x ∂ P z ∂ E 0 0 − Ω P T E tan λ ∂ tan λ ∂ tan λ ∂ tan λ ∂ tan λ         P 2 ∂ P y ∂ P x ∂ P z ∂ E   P x P y P z     ∂ Ω ∂ Ω ∂ Ω ∂ Ω E           = − 1 ∂ P y   ∂ P x ∂ P z 0 0 0 0 ∂ E   J =    ∂ d 0 ∂ d 0 ∂ d 0 ∂ d 0  Ω           0 0 0 0  ∂ P y  ∂ P x ∂ P z ∂ E     ∂ z 0 ∂ z 0 ∂ z 0 ∂ z 0           P y Ω − P x Ω 0 0   ∂ P y   ∂ P x ∂ P z ∂ E ∂φ ∂φ ∂φ ∂φ → Σ ′ i = J T Σ i J , covariance matrix in momenta space. i = J Σ i J T if you define jacobian as the transposed of quoted above) ( Σ ′ C. Calancha (KEK) Covariance Matrix April 16, 2014 11/11