Kinematic fitting. A powerful tool of event selection and reconstruction. Sergey Yaschenko, DESY Zeuthen. University of Glasgow,

Transcription

1 Knematc fttng A owerful tool of event selecton and reconstructon Serge Yaschenko, DESY Zeuthen Unverst of Glasgow,..

2 Outlne > Introducton > Technques > Alcatons > Results of use at HERMES > Summar > Lterature Serge Yaschenko Knematc fttng.. Page

3 Introducton: knematc fttng as a art of data rocessng > Raw data rocessng > Detector calbratons > Track search and reconstructon > Momentum reconstructon > Event selecton and reconstructon At ths stage -momenta of all found tracks are reconstructed Combne ndvdual tracks to events Al certan requrements (cuts) on track correlatons to select events of nterest and reject the background Alternatvel use knematc fttng > Phscs analss Serge Yaschenko Knematc fttng.. Page

4 Introducton: knematc fttng as a tool of event selecton > Knematc fttng adjustment of measured knematc arameters under certan assumtons (condtons, constrants) > Ams Test f assumtons are true Imrove accurac of measurements Check f the knowledge of measurement uncertantes s correct Check for ossble sstematc uncertantes > Has been used for more than 5 ears n artcle hscs, consdered as a standard tool of event selecton and reconstructon > But also often consdered as too comlcated and not reall necessar Serge Yaschenko Knematc fttng.. Page 4

5 Examle > Deca of a artcle a to artcles a ++ > Energ of the rmar artcle s recsel known and equal to E a = GeV > Energes of the secondar artcles are equal to E = 5 GeV, E = GeV and E = 5 GeV and measured wth errors dstrbuted b Gaussan wth sgmas σ = GeV, σ = GeV and σ = GeV > Mnmzaton of least-squares functonal > under condton E ft ( E E ) / ft ft ft E E Ea > If measurement errors are the same the result s trval: ft E E ( E E E Ea ) / Serge Yaschenko Knematc fttng.. Page 5

6 Results of examle > Ch square dstrbuted as ch square for one degree of freedom > Probablt dstrbuton > A measure of the robablt that a ch square from the theoretcal dstrbuton s greater than ch square obtaned from the ft f ( z; n) dz > Imrovement of accuraces of energ measurements b factor of (/) χ Probablt Energ [GeV] Serge Yaschenko Knematc fttng.. Page 6

7 Background rocess > Deca of a artcle a to 4 artcles a +++4 > Energ of the rmar artcle s recsel known and equal to E a = GeV > Energes of the secondar artcles are equal to E = GeV, E = 8 GeV and E = GeV and measured wth errors dstrbuted b Gaussan wth sgma σ = GeV, σ = GeV and σ = GeV > Partcles 4 wth energ of 6 GeV s unmeasured (mssed artcle) > Fttng assumng a ++ hothess χ Probablt Energ [GeV] Serge Yaschenko Knematc fttng.. Page 7

8 Mssng energ method vs knematc fttng > Process of nterest: mssng energ s eaked near zero, ch square s dstrbuted as ch square for degree of freedom Mssng Energ [GeV] > Background rocess: mssng energ eak s shfted, ch square dstrbuton s comletel dfferent Mssng Energ [GeV] χ χ Serge Yaschenko Knematc fttng.. Page 8

9 Mssng energ method vs knematc fttng > Al a cut on the mssng energ or to ch-square or robablt dstrbuton to select the rocess of nterest and reject the background Mssng Energ [GeV] Probablt Mssng Energ [GeV] Probablt Serge Yaschenko Knematc fttng.. Page 9

10 Knematc fttng technque > Mnmzaton of least squares functonal under constrants n ( ) / f(,,..., n ) f(,,..., n ) fm (,,..., n) measured knematc arameters, η ft arameters, σ measurement errors, n number of knematc arameters, m number of constrants > In case of correlatons between knematc arameters, covarance matrx G should be used n n j ( ) G ( ) (, j) j j Serge Yaschenko Knematc fttng.. Page

11 Mnmzaton of least-squares functons wth constrants > Method of elements Usng q equatons of constrants elmnate q of the n knematc arameters as functons of n-q arameters Not alwas ossble, rocedure s not automatc > Method of Lagrange multles Automatc rocedure, wdel used Exact soluton f constrants deend lnearl on arameters Smle teratve rocedure f constrants are non-lnear > Method of orthogonal transformaton Mentoned n one textbook, not wdel used > Method of enalt functons Automatc rocedure Can be effectvel used wth constrants of nequalt te Serge Yaschenko Knematc fttng.. Page

12 Method of Lagrange multlers > Measurements gve nstead of true quanttes η values > Measurement errors are normall dstrbuted about zero wth standard devatons σ > m equatons of constrants > If equatons are lnear wrte n the matrx form > Defne E( ) E( ), f k, k,,... m B b c B b > Mnmzaton of Lagrange functon L T G T ( c B ) > Total dervatve should vansh dl T T G d Bd Serge Yaschenko Knematc fttng.. Page

13 Method of Lagrange multlers > Solve sstem of lnear equatons > Obtan > Can be easl solved > Estmators of the measurement errors > Estmators for ft arameters > Usng abbrevaton > Obtan b alng error roagaton T T G B c B T c BG B ~ BG B T c ~ T G B T BG B c ~ ~ T G B BG B G~ G G B T G B BG Serge Yaschenko Knematc fttng.. Page

14 Serge Yaschenko Knematc fttng.. Page 4 Soluton of examle > In the examle > Vector > Estmaton of ft arameters > Estmaton of error matrx of ft arameters G~ B G B ) ( ~ a E E E E E a a E E E E E E E E c

15 Examle > Deca of a artcle a to artcles a ++ > Energ of the rmar artcle s recsel known and equal to E a = GeV > Energes of the secondar artcles are equal to E = 5 GeV, E = GeV and E = 5 GeV and measured wth errors dstrbuted b Gaussan wth sgma σ = GeV, σ = GeV and σ = 5 GeV χ Energ [GeV] E E ft ( E E E E a ) /( ) Probablt Serge Yaschenko Knematc fttng.. Page 5

16 Results of examle > Pull dstrbutons defned as ull ( ) ( ) ( ) > Should be normall dstrbuted around zero wth standard devaton equal to unt > Hel to understand f measurement errors are known correctl > Check for ossble sstematc effects χ / ndf / 8 Constant 98 ± 5.5 Mean -.99 ±.7 Sgma. ± Pull χ / ndf / 8 Constant 98 ± 5.5 Mean -.99 ±.7 Sgma. ± χ / ndf 75.8 / 79 Constant 4 ± 6. Mean -.47 ±.54 Sgma.9649 ± Pull Pull Serge Yaschenko Knematc fttng.. Page 6

17 Examle > Deca of a artcle artcles a ++ > Energ of the rmar artcle s recsel known and equal to E a = GeV > Energes of the secondar artcles are equal to E = 5 GeV, E = GeV and E = 5 GeV and measured wth errors dstrbuted b Gaussan wth sgma σ = GeV, σ = GeV and σ = 5 GeV > Measurement of error of E s not known and assumed to be GeV χ Probablt Energ [GeV] Serge Yaschenko Knematc fttng.. Page 7

18 Results of examle > Pull dstrbutons are wder, ndcaton that somethng s wrong wth the knowledge of measurement errors χ / ndf 8.6 / 66 Constant 878 ± 7. Mean ±.67 Sgma. ± Pull Pull χ / ndf 8.6 / 66 Constant 878 ± 7. Mean ±.67 Sgma. ± Pull χ / ndf 6. / 4 Constant ± 8.6 Mean ±.57 Sgma.89 ±.4 Serge Yaschenko Knematc fttng.. Page 8

19 Examle 4 > Deca of a artcle artcles a ++ > Energ of the rmar artcle s recsel known and equal to E a = GeV > Energes of the secondar artcles are equal to E = 5 GeV, E = GeV and E = 5 GeV and measured wth errors dstrbuted b Gaussan wth sgma σ = GeV, σ = GeV and σ = 5 GeV > Bas n the measurement of E : 7 GeV nstead of 5 GeV χ Probablt Energ [GeV] Serge Yaschenko Knematc fttng.. Page 9

20 Results of examle 4 > Pull dstrbutons are shfted χ / ndf / 8 Constant 98 ± 5.5 Mean.88 ±. Sgma. ± Pull χ / ndf / 8 Constant 98 ± 5.5 Mean.88 ±. Sgma. ± Pull χ / ndf 8.5 / 78 Constant 44 ± 6. Mean.668 ±. Sgma.964 ± Pull Serge Yaschenko Knematc fttng.. Page

21 Alcatons of knematc fttng > Event selecton > Background rejecton > Imrovement of resoluton > Better understandng of measurement errors, search for ossble sources of sstematc uncertantes Analss of ch square (robablt) and ull dstrbutons > Detector calbraton If number of constrants s enough, some measurements can be consdered as unknown arameters, reconstructed b knematc fttng and then used for calbraton Serge Yaschenko Knematc fttng.. Page

22 Possble roblems > Measurement errors are not known recsel > Sstematc uncertantes > Measurement errors are not Gaussan > If dfferent artcles are measured n dfferent coordnate sstems msalgnment between these sstems > Multle scatterng and radatve effects usuall lead to tals n chsquare and ull dstrbutons Serge Yaschenko Knematc fttng.. Page

23 Outlne > Introducton > Technques > Alcatons > Results of use at HERMES > Lterature > Summar Serge Yaschenko Knematc fttng.. Page

24 Deel vrtual Comton scatterng (DVCS) at HERMES DVCS e e e Bethe-Hetler e e e γ* γ γ γ x+ ξ x-ξ γ* γ* GPDs(x,ξ,t) t > DVCS and Bethe-Hetler: the same ntal and fnal state > Bethe-Hetler domnates at HERMES knematcs > Generalzed Parton Dstrbutons (GPDs) accessble through cross secton dfferences and azmuthal asmmetres va nterference term Serge Yaschenko Knematc fttng.. Page 4

25 The HERMES exerment m TARGET CELL SILICON ' FRONT MUON HODO DRIFT CHAMBERS DVC HODOSCOPE H FC / PROP. CHAMBERS MC FIELD CLAMPS DRIFT CHAMBERS BC / TRIGGER HODOSCOPE H 7 mrad PRESHOWER (H) LUMINOSITY MONITOR e' 7 mrad e+ STEEL PLATE BC /4 TRD CALORIMETER 4 mrad 7.5 GeV 4 mrad RICH 7 mrad IRON WALL 7 mrad MAGNET WIDE ANGLE MUON HODOSCOPE MUON HODOSCOPES Gas targets: Longtudnall olarzed H, D Unolarzed H, D, 4 He, N, Ne, Kr, Xe Transversel olarzed H Beam: Longtudnall olarzed e + and e wth both helctes Energ 7.6 GeV Serge Yaschenko Knematc fttng.. Page 5 m

26 DVCS/BH measurement wth the Recol Detector N/N DIS. e + data - e data. e MC sum elastc BH assocated BH sem-nclusve > Pre-Recol data Scattered leton and hoton were detected n the forward sectrometer Recol roton was not detected Exclusvt acheved va mssng mass technque Assocated rocesses (ee γ) were not resolved (% contrbuton n the sgnal). Tesla suerconductng solenod (GeV ) > Recol data Detecton of recol roton Suresson of the background to <% level M x Photon Detector (PD) Scntllatng Fber Tracker Leton beam Slcon Str Detector Target cell of unolarzed target Serge Yaschenko Knematc fttng.. Page 6

27 Elastc DVCS/BH and assocated background > Elastc DVCS/BH: e eγ Beam -momentum s known recsel (n comarson wth -momenta of secondar artcles) Target roton s at rest For e,, and γ, -momenta are measured wth certan recson > Background from assocated rocess ee + γ wth + π > Not ossble to searate b mssng mass onl > Low effcenc of selecton f one-dmensonal cuts (colanart, momentum-momentum correlatons) are aled Serge Yaschenko Knematc fttng.. Page 7

28 Selecton of elastc DVCS/BH usng smle cuts > One-dmensonal cuts Momentum-momentum correlatons Colanart condton > Effcenc of 7% at background contamnaton of 5% > background contamnaton of % can be acheved but effcenc dros below 5% > No mrovement of resolutons counts 5 5 Hermes 7 data Tradtonal DVCS analss Recol roton n accetance (E γ >5 GeV) wth Colanart cut Colanart cut turned around M x [ (GeV/c) ] Serge Yaschenko Knematc fttng.. Page 8

29 Serge Yaschenko Knematc fttng.. Page 9 Knematc fttng for elastc DVCS/BH > Nne knematc arameters: Scattered electron Photon Proton > Four equatons of constrants / ) / tan( ) / tan( z z x ) sn /( beam beam z z z x x x m e e e e f f f f ) / tan( ) / tan( e z z x

30 Serge Yaschenko Knematc fttng.. Page Constrants as functons of arameters > Constrants are non-lnear > Dervatves of constrants can be calculated analtcall or numercall beam e beam m e m m f f f f )) ( sn /( / )) tan( /( / / / ) / sn( / / / ) / cos( / / /

31 Serge Yaschenko Knematc fttng.. Page Fttng rocedure > Mnmzaton of ch square wth constrants usng enalt term > If T s large enough constrants are satsfed automatcall durng the fttng rocedure > In order to equalze enalt contrbutons from dfferent constrants errors of constrants are formall calculated usng error roagaton > Non-lnear constrants of nequalt te can be ncluded m k ck k n j j j n j f T G ), ( / ) ( ) (

32 Results for elastc DVCS/BH and assocated background > For elastc DVCS/BH (Monte Carlo) χ > For assocated background Probabllt χ χ Probabllt Serge Yaschenko Knematc fttng.. Page

33 Effcenc of elastc event selecton Effcenc χ cut Background contamnaton χ cut > Effcenc of event selecton of elastc DVCS/BH events s hgh at a reasonable cut on ch-square > Contamnaton of assocated rocess s well below % > In addton, mrovement of resolutons Serge Yaschenko Knematc fttng.. Page

34 Imrovement of knematc arameter reconstructon Momentum of electron Energ of hoton Δ /.5. Δ / Momentum [GeV/c] Polar angle of roton Momentum [GeV/c] Momentum of roton Δ θ [rad] Δ / Momentum [GeV/c] Momentum [GeV/c] Serge Yaschenko Knematc fttng.. Page 4

35 t resoluton > Mandelstam t=(- ), mortant hscal observable > Blue squares reconstructon usng measured knematc arameters of electron and hoton > Magenta trangles - reconstructon usng measured knematc arameters of recol roton ] Δ t [GeV > Lght blue trangles - reconstructon usng measured knematc arameters of electron and hoton (excludng hoton energ) assumng roton mass t [GeV ] > Red squares reconstructon usng knematc fttng Serge Yaschenko Knematc fttng.. Page 5

36 Pull dstrbutons Electron Photon Proton χ / ndf 6.9 / 9 Constant 7.7 ±.4 Mean ±.789 Sgma.7 ± Pull χ / ndf.4 / 8 Constant 67.8 ±.4 Mean -.8 ±.745 Sgma.6 ± Pull χ / ndf 87. / 5 Constant 78.7 ±.5 Mean ±.7 Sgma.986 ± Pull χ / ndf 44.6 / Constant 7. ±.5 Mean ±.77 Sgma. ± Pull χ / ndf 4.6 / Constant 76. ±.5 Mean ±.788 Sgma.99 ± Pull χ / ndf. / 4 Constant 69.5 ±.5 Mean.69 ±.746 Sgma. ± Pull χ / ndf 49. / 4 Constant 6 ±.5 Mean -.69 ±.776 Sgma. ± Pull χ / ndf 8.9 / 7 Constant 69.7 ±.4 Mean.8947 ±.767 Sgma. ± Pull χ / ndf 9.6 / 9 Constant 67.9 ±.6 Mean -.8 ±.74 Sgma.9 ± Pull 9 Serge Yaschenko Knematc fttng.. Page 6

37 Selecton of elastc DVCS/BH events (HERMES data) > Knematc fttng s develoed and tested on Monte-Carlo artcles detected 4 constrants from energ-momentum conservaton Allows to suress the assocated rocesses and sem-nclusve background to neglgble level > Aled for data for hscs analss Sstematc studes n rogress Frst hscs results exected soon > Mssng mass dstrbuton No requrement for Recol Postvel charged Recol track Knematc ft robablt > % Knematc ft robablt < % Data MX [GeV /c ] Serge Yaschenko Knematc fttng.. Page 7

38 Summar > Knematc fttng s a owerful tool of event selecton and reconstructon > Man alcatons Imrovement of resoluton Better understandng of measurement errors, search for ossble sources of sstematc uncertantes Detector calbraton > Knematc fttng technque s well develoed and descrbed n lterature > Dfferent roblems could aear n dfferent exerments, solutons are not alwas straghtforward Serge Yaschenko Knematc fttng.. Page 8

39 Lterature (ersonal choce) > W.E.Demng, Statstcal Adjustment of Data > S.Brandt, Data Analss > R.Frühwrth, M.Regler, R.K.Bock, H. Grote, D. Notz, Data Analss Technques for Hgh-Energ Phscs > J.P.Berge, F.T.Solmtz, H.D.Taft, Rev. Scent. Instr. N5 (96) 58 > R.Boeck CERN 6- > W.G.Davs, Nucl. Instr. Meth A 86 (997) 498 > S.N.Dmov et al, Nucl. Instr. Meth A 44 () 4 > M.Wllams, C.A.Meer, CLAS-NOTE -7 > P.Aver, htt:// > V.Bloebel, htt:// Serge Yaschenko Knematc fttng.. Page 9