1 st G lobal QCD Analysis of Polarized Parton Densities

1. October 7th, 2008 1st Global QCD Analysis of Polarized Parton Densities Marco Stratmann

2. work done in collaboration with Daniel de Florian(Buenos Aires) Rodolfo Sassot(Buenos Aires) Werner Vogelsang (BNL) references • Global analysis of helicity parton densities and their uncertainties, PRL101 (2008) 072001 (arXiv:0804.0422 [hep-ph]) • a long, detailled paper focussing on uncertainties is in preparation DSSV pdfs and further information available from ribf.riken.jp/~marco/DSSV

3. the challenge: analyze a large body of data from many experiments on different processes with diverse characteristics and errors within a theoretical model with many parameters and hard to quantify uncertainties without knowing the optimum “ansatz” a priori

4. information on nucleon spin structure available from • each reaction provides insights into different aspects and x-ranges • all processes tied together: universality of pdfs & Q2 - evolution • need to use NLO task: extract reliable pdfs not just compare some curves to data

5. details & results of • the DSSV global analysis • toolbox • comparison with data • uncertainties from Lagrange multipliers • comparison with Hessian method • next steps

6. QCD scale evolution due to resolving more and moreparton-parton splittings as the “resolution” scale mincreases the relevant DGLAP evolution kernels are known to NLO accuracy: Mertig, van Neerven; Vogelsang m - dependence of PDFs is a key prediction of pQCD verifying it is one of the goals of a global analysis 1. theory “toolbox”

7. Jäger,MS,Vogelsang • higher order corrections essential to estimate/control theoretical uncertainties closer to experiment (jets,…) scale uncertainty all relevant observables available at NLO accuracy except for hadron-pair production at COMPASS, HERMES Q2' 0 available very soon:Hendlmeier, MS, Schafer • factorization e.g., pp !p X allows to separate universal PDFs from calculable but process-dependent hard scatterring cross sections

8. “classic” inclusive DIS data routinely used in PDF fits !Dq + Dq semi-inclusive DIS data so far only used in DNS fit !flavor separation first RHIC pp data (never used before) !Dg 2. “data selection” initial step: verify that the theoretical framework is adequate ! !use only data where unpolarized results agree with NLO pQCD DSSV global analysis uses all three sources of data: 467 data pts in total (¼10% from RHIC)

9. DSS analysis:(de Florian, Sassot, MS) • first global fit of FFs including e+e-, ep, and pp data • describe all RHIC cross sections and HERMES SIDIS multiplicities • (other FFs (KKP, Kretzer) do not reproduce, e.g., HERMES data) • uncertainties on FFs from robust Lagrange multiplier method and propagated to DSSV PDF fit ! details: • Global analysis of fragmentation functions for pions and kaons • and their uncertainties, Phys. Rev. D75 (2007) 114010(hep-ph/0703242) • Global analysis of fragmentation functions for protons • and charged hadrons, Phys. Rev. D76 (2007) 074033(arXiv:0707.1506 [hep-ph]) data with observed hadrons • SIDIS(HERMES, COMPASS, SMC) • pp !p X(PHENIX) strongly rely on fragmentation functions !new DSS FFs are a crucial input to the DSSV PDF fit Fortran codes of the DSS fragmentation fcts are available upon request

10. take as from MRST; also use MRST for positivity bounds • NLO fit, MS scheme • avoid assumptions on parameters {aj} unless data cannot discriminate need to impose: let the fit decide about F,D value constraint on 1st moments: 1.269§0.003 fitted 0.586§0.031 3. setup of DSSV analysis • flexible, MRST-like input form possible nodes input scale simplified form for sea quarks and Dg: kj = 0

11. 4. fit procedure change O(20) parameters {aj} about 5000 times another 50000+ calls for studies of uncertainties 467 data pts bottleneck ! computing time for a global analysis at NLO becomes excessive problem: NLO expression for pp observables are very complicated

12. well-known property: convolutions factorize into simple products • analytic solution of DGLAP evolution equations for moments • analytic expressions for DIS and SIDIS coefficient functions !problem can be solved with the help of 19th century math idea: take Mellin n-moments R.H. Mellin Finnish mathematician inverse … however, NLO expression for pp processes too complicated

13. MS, Vogelsang completely indep. of pdfs pre-calculate prior to fit express pdfs by their Mellin inverses discretize on 64 £ 64 grid for fast Gaussian integration standard Mellin inverse fit here is how it works: earlier ideas: Berger, Graudenz, Hampel, Vogt; Kosower example: pp!p X

14. applicability & performance • obtaining the grids once prior to the fit 64 £ 64 £ 4 £ 10 ' O(105) calls per pp data pt. n m # subproc’s n,m complex production of grids much improved recently can be all done within a day with new MC sampling techniques • method completely general tested for pp!gX, pp!pX, pp!jetX (much progress towards 2-jet production expected from STAR) • computing load O(10 sec)/data pt.!O(1 msec)/data pt. recall: need thousands of calls to perform a single fit !

15. details & results of • the DSSV global analysis • toolbox • comparison with data • uncertainties from Lagrange multipliers • comparison with Hessian method • next steps

16. c2/d.o.f. ' 0.88 note: for the time being, stat. and syst. errors are added in quadrature overall quality of the global fit very good! no significant tension among different data sets

17. new data sets used in: the old GRSV analysis the combined DIS/SIDIS fit of DNS inclusive DIS data

18. we only account for the “kinematical mismatch” between A1 and g1/F1 in (relevant mainly for JLab data) • very restrictive functional form in LSS: Df = N ¢ x a¢ fMRST only 6 parameters for pdfs but 10 for HT • very limited Q2 – range ! cannot really distinguish ln Q2 from 1/Q2 • relevance of CLAS data “inflated” in LSS analysis: 633 data pts. in LSSvs. 20 data pts. in DSSV in a perfect world this should not matter, but … remark on higher twist corrections • no need for additional higher twist corrections (like in Blumlein & Bottcher) at variance with results of LSS (Leader, Sidorov, Stamenov) – why?

19. semi-inclusive DIS data not in DNS analysis impact of new FFs noticeable!

20. RHIC pp data (inclusive p0 or jet) • good agreement • important constraint on Dg(x) despite large uncertainties ! later uncertainty bands estimated with Lagrange multipliers by enforcing other values for ALL

21. details & results of • the DSSV global analysis • toolbox • comparison with data • uncertainties from Lagrange multipliers • comparison with Hessian method • next steps

22. track c2 • finds largest DOiallowed by the global data set • and theoretical framework for a given Dc2 • explores the full parameter space {aj} independent of approximations Lagrange multiplier method see how fit deteriorates when PDFs are forced to give a different prediction for observable Oi • Oican be anything: we have looked at ALL, truncated 1st moments, • and selected fit parameters aj so far • requires large series of minimizations (not an issue with fast Mellin technique)

23. !idealistic Dc2=1 $ 1sapproach usually fails we present uncertainties bands for both Dc2 = 1 and a more pragmatic 2% increase in c2 also: • Dc2 = 1 defines 1s uncertainty for single parameters • Dc2 ' Nparis the 1s uncertainty for allNpar parameters • to be simultaneously located in “c2-hypercontour” used by AAC see: CTEQ, MRST, … Dc2 - a question of tolerance What value of Dc2 defines a reasonable error on the PDFs ? certainly a debatable/controversial issue … • combining a large number of diverse exp. and theor. inputs • theor. errors are correlated and by definition poorly known • in unpol. global fits data sets are marginally compatible at Dc2 = 1

24. summary of DSSV distributions: • robust pattern of flavor-asymmetric • light quark-sea (even within uncertainties) • Du + Du and Dd + Dd very similar to GRSV/DNS results • small Dg, perhaps with a node • Ds positive at large x • Du > 0, Dd < 0 predicted in some models Diakonov et al.; Goeke et al.; Gluck, Reya; Bourrely, Soffer, …

25. Dc2 Dc2 x • determined by SIDIS data • mainly charged hadrons • pions consistent a closer look at Du • small, mainly positive • negative at large x

26. Dc2 Dc2 x • determined by SIDIS data • DIS alone: more negative • mainly from kaons, a little bit from pions a closer look at Ds striking result! • positive at large x • negative at small x

27. study uncertainties in 3 x-regions find • Dg(x) very small at medium x (even compared to GRSV or DNS) • best fit has a node at x ' 0.1 • huge uncertainties at small x x small-x 0.001· x · 0.05 RHIC range 0.05· x · 0.2 large-x x ¸ 0.2 a closer look at Dg error estimates more delicate: small-x behavior completely unconstrained

28. Ds receives a large negative contribution at small x • Dg: huge uncertainties below x'0.01 !1st moment still undetermined • eSU(2),SU(3) come out close • to zero eSU(2) eSU(3) 1st moments: Q2 = 10 GeV2

29. details & results of • the DSSV global analysis • toolbox • comparison with data • uncertainties from Lagrange multipliers • comparison with Hessian method • next steps

30. displacement: Hessian Hij taken at minimum only quadratic approximation J. Pumplin et al., PRD65(2001)014011 can benefit from a lot of pioneering work by CTEQand use their improved iterative algorithm to compute Hij good news: Hessian method estimates uncertainties by exploring c2 near minimum: • easy to use (implemented in MINUIT) but not necessarily very robust • Hessian matrix difficult to compute with sufficient accuracy in complex problems like PDF fits where eigenvalues span a huge range

31. sets Sk§ can be used to calculate uncertainties of observables Oi PDF eigenvector basis sets SK§ cartoon by CTEQ • eigenvectors provide an optimized orthonormal basis • to parametrize PDFs near the global minimum • construct 2Npar eigenvector basis sets Sk§ by displacing each zk by § 1 • the “coordinates” are rescaled such that Dc2 = åk zk2

32. tend to be a bit larger • for Hessian, in particular • for Dg(x) • uncertainties of truncated moments for Dc2=1 agree well except for Dg comparison with uncertainties from Lagrange multipliers • Hessian method goes • crazy if asking for Dc2>1

33. details & results of • the DSSV global analysis • toolbox • comparison with data • uncertainties from Lagrange multipliers • comparison with Hessian method • next steps

34. 1. getting ready to analyze new types of data from the next long RHIC spin run with O(50pb-1) and 60% polarization • significantly improve existing • inclusive jet + p0 data • (plus p+, p-, …) • first di-jet data from STAR • ! more precisely map Dg(x) the Mellin technique is basically in place to analyze also particle correlations challenge: much slower MC-type codes in NLO than for 1-incl. from 2008 RHIC spin plan

35. 2. further improving on uncertainties • Lagrange multipliers more reliable than Hessian with present data • Hessian method perhaps useful for Dc2 = 1 studies, beyond ?? • include experimental error correlations if available planning ahead: at 500GeV the W-boson program starts • flavor separation independent of SIDIS ! important x-check of present knowledge • implementation in global analysis (Mellin technique) still needs to be done available NLO codes (RHICBos) perhaps too bulky • would be interesting to study impact with some simulated data soon

37. extra slides

38. DSS: good global fit of all e+e-andep, pp data de Florian, Sassot, MS main features: • handle on gluon fragmentation • flavor separation • uncertainties via Lagrange multipl. • results for p§, K§, chg. hadrons

39. meet the distributions: Dd Dc2 Dc2 x • fairly large • negative throughout • determined by SIDIS data • some tension between charged hadrons and pions

40. significant deviations from assumed quadratic dependence c2 profiles of eigenvector directions #1: largest eigenvector (steep direction in c2) … #19: smallest eigenvector (shallow direction in c2) for a somewhat simplified DSSV fit with 19 parameters

41. look O.K. but not necessarily parabolic Dg mixed bag steep shallow worse for fit parameters: mix with all e.v. (steep & shallow)

42. roughly corresponds to what we get from Lagrange multipliers the good … … the bad … the ugly