Momentum Resolution Event-by-Event Basis C. Calancha - PowerPoint PPT Presentation

Momentum Resolution Event-by-Event Basis C. Calancha calancha@post.kek.jp 2014, April 4 C. Calancha (KEK) Momentum Resolution 2014, April 4 1

Momentum Resolution Event-by-Event Basis Developped code for covariance matrix P momenta Last meetings i reported i was developping this code. See my previous talks for more details. I have written marlin processor adding new LCCollection of pfos with the cov. matrix filled. I have repeated my calculations. During that calculations some questions arise to my mind. I found answer to such questions. Today i talk about and share what i have learned about it (*). (*) If you are not interested in linear algebra please jump to summary. C. Calancha (KEK) Momentum Resolution 2014, April 4 2

Understanding the new variable Change of Basis We are actually doing a change of basis: Original base: A = { tan λ , Ω , φ , d 0 , z 0 } New base: B = { p x , p y , p z , E } Original basis has higher rank: full description of the phenomena. B is actually expanding just one subspace of the total space H expanded by A . That looks logical: with A provides position and momenta of the track. B just provide momenta/energy. Could correlations of basis vectors { tan λ , Ω , φ } with { d 0 , z 0 } have impact on correlations of B basis vectors. Say in other words: Is it the subspace expanded by { d 0 , z 0 } orthogonal to the subspace generated by { p x , p y , p z , E } ? Or alternatively: Should i use the full covariance matrix in helicity parameters space ( A ) when traslating it to the new basis ( B )? My first thought was saying: no, i dont need it. Then i saw other experiments use my same expressions (CDF, LHCb). But, is it just a valid aproximation? Is it exact? I want to know it. Good opportunity to learn something new. C. Calancha (KEK) Momentum Resolution 2014, April 4 3

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    E  ∂ Ω ∂ Ω ∂ Ω ∂ Ω           = − 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 x Ω P y 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) Momentum Resolution 2014, April 4 4

Is there effect on d0 z0 on the Covariance matrix? 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 Σ ′ 1 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 Should i include full matrix (rank 5) in the item 1)? The goal of this report is to answer this question. C. Calancha (KEK) Momentum Resolution 2014, April 4 5

Is there effect on d0 z0 on the Covariance matrix? Momenta does not depend on d 0 , z 0 p x = p x ( tan λ , Ω , φ ) p y = p y ( tan λ , Ω , φ ) p z = p z ( tan λ , Ω) Intuitively space/momentum are independent measurements, but... are they? Position and momenta info. comming as a result of track fitting. So, eventually they are not independent measurements: the info comes from same fits. The covariance in helix parameters comes from track fitting. The covariance in A space is a symmetric 5x5 matrix with (generally) non null elements. That means, every variable has some correlation with others: Cov ( i , i ) � = 0 , ∀ i , j In particular, d 0 or z 0 correlation on tan λ . So, as p x depend on tan λ , why not d 0 , z 0 effect on, for instance tan λ , be translated to p x when we go from A to B ? For me it is not obvious why those correlations should canceled. C. Calancha (KEK) Momentum Resolution 2014, April 4 6

Same Result using full matrix or not Comparison cov. matrix using 3x3 and 5x5 helix matrix cov xx cov yx cov yy cov zx cov zy cov zz ... cov 3x3 1.57576e-05 -3.9027e-06 2.10397e-06 2.88209e-05 -7.63759e-06 5.71544e-05 ... cov 5x5 1.57576e-05 -3.9027e-06 2.10397e-06 2.88209e-05 -7.63759e-06 5.71544e-05 ... The covariance matrix is exactly the same. That means d 0 , z 0 correlations canceled identically. C. Calancha (KEK) Momentum Resolution 2014, April 4 7

Cancelation Proof I To show this cancelation is useful to order the base vectors in the following way: A = { d 0 , z 0 , tan λ , Ω , φ } B = { p x , p y , p z , E } Now, the jacobian looks like (first two rows are null): ∂ P y 2 ∂ P x ∂ P z ∂ E 3 2 0 0 0 0 3 ∂ d 0 ∂ d 0 ∂ d 0 ∂ d 0 6 7 6 7 6 7 6 0 0 0 0 7 ∂ P y 6 ∂ P x ∂ P z ∂ E 7 6 7 6 7 6 7 ∂ z 0 ∂ z 0 ∂ z 0 ∂ z 0 6 7 6 7 − P 2 6 7 6 z Ω 7 = − 1 − Ω P T 6 7 6 0 0 7 ∂ P y J = ∂ P x ∂ P z ∂ E E tan λ 6 7 6 7 6 7 Ω 6 7 ∂ tan λ ∂ tan λ ∂ tan λ ∂ tan λ 6 7 6 7 P 2 6 7 6 7 P x P y P z 6 ∂ P y 7 6 7 ∂ P x ∂ P z ∂ E E 6 7 6 7 ∂ Ω ∂ Ω ∂ Ω ∂ Ω 6 7 6 7 6 7 6 7 P y Ω − P x Ω 0 0 4 5 4 5 ∂ P y ∂ P x ∂ P z ∂ E ∂φ ∂φ ∂φ ∂φ C. Calancha (KEK) Momentum Resolution 2014, April 4 8

The cov. matrix in the original basis looks like: Cov [ d 0 , d 0 ] Cov [ d 0 , z 0 ] Cov [ d 0 , tan λ ] Cov [ d 0 , Ω] Cov [ d 0 , φ ] 2 3 6 7 Cov [ z 0 , d 0 ] Cov [ z 0 , z 0 ] Cov [ z 0 , tan λ ] Cov [ z 0 , Ω] Cov [ z 0 , φ ] 6 7 6 7 6 7 6 7 Cov [ tan λ, d 0 ] Cov [ tan λ, z 0 ] Cov [ tan λ, tan λ ] Cov [ tan λ, Ω] Cov [ tan λ, φ ] 6 7 A = 6 7 6 7 6 7 Cov [Ω , d 0 ] Cov [Ω , z 0 ] Cov [Ω , tan λ ] Cov [Ω , Ω] Cov [Ω , φ ] 6 7 6 7 6 7 6 7 Cov [ φ, d 0 ] Cov [ φ, z 0 ] Cov [ φ, tan λ ] Cov [ φ, Ω] Cov [ φ, φ ] 4 5 C. Calancha (KEK) Momentum Resolution 2014, April 4 9

Cancelation Proof II i = J T Σ i J Σ ′ A = { d 0 , z 0 , tan λ , Ω , φ } Σ ′ i = ( b ij ) Σ i = ( a ij ) B = { p x , p y , p z , E } J = ( h ij ) b ij = P 5 P 5 m = 1 h ri a rm · h mj r = 1 , = P 5 m = 1 ( P 5 r = 1 h ri a rm ) · h mj , = P 5 m = 1 ( 0 · a 1 m + 0 · a 2 m + P 5 r = 3 h ri a rm ) · h mj , = () · h 1 j + () · h 2 j + P 5 m = 3 ( 0 · a 1 m + 0 · a 2 m + P 5 r = 3 h ri a rm ) · h mj , = P 5 m = 3 ( P 5 r = 3 h ri a rm ) · h mj ( a ij ) elements related with d 0 , z 0 does not contribute to ( b ij ) (see previous slide). Geometrically that means: subspaces generated by { d 0 , z 0 } and B are orthogonal. From an experimental point of view: i dont need to use the full (5x5) covariance matrix in helicity parameters (just the 3x3). C. Calancha (KEK) Momentum Resolution 2014, April 4 10

Marlin Processor I have written a new Marlin Processor filling the covariance matrix in P . Output is a new LCCollection copy of PandoraPFOs but with non null cov. matrix. This code should be included in new releases of ILCSOFT . Code will be available very soon (hopefully this evening) at kekcc here: /hsm/ilc/grid/JB/users/calancha/code/marlin/momentumCov Example and xmlfile will be provided in same directory as well. C. Calancha (KEK) Momentum Resolution 2014, April 4 11

Summary Developed piece of code to get covariance matrix of charged particles in momenta space. Understood why i just need { tan λ , Ω , φ } variables in the calculation. A new LCCollection is added to the event (copy of PandoraPFOs) with filled cov. matrix. Used it to calculate dimuon mass error event-by-event, but it is useful on its own (directly related with ILD tracking). C. Calancha (KEK) Momentum Resolution 2014, April 4 12

BACKUP BACK UP C. Calancha (KEK) Momentum Resolution 2014, April 4 13

Last meeting a show some scatter plots with the new variable. Plots were not clear due to not right choice of the axis. Its more clear to plot profile plots to see the dependence of the two variables. C. Calancha (KEK) Momentum Resolution 2014, April 4 14

ILD Preliminary ILD Preliminary 3 ] 2 )) [GeV/c 2 µ , µ (M( 1 σ 0 -1 -0.5 0 0.5 1 θ cos( ) H Better precision at central region (tracks have more hits). C. Calancha (KEK) Momentum Resolution 2014, April 4 15

No dependence on azimutal angle. C. Calancha (KEK) Momentum Resolution 2014, April 4 16

Gaussian Fit [-2,2] ILD Preliminary µ 600 fit : -0.157 σ fit : 1.00013 400 200 0 -20 -10 0 10 20 µ µ σ µ µ (M( , ) - 125 ) / (M( , )) C. Calancha (KEK) Momentum Resolution 2014, April 4 17