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

2. 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, PRL 101 (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 ribf.riken.jp/~marco/DSSV

3. 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. 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 & Q 2 - evolution  need to use NLO task: extract reliable pdfs not just compare some curves to data

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

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

7. 7  factorization allows to separate universal PDFs from calculable but process-dependent hard scatterring cross sections e.g., pp !  X  higher order corrections essential to estimate/control theoretical uncertainties closer to experiment (jets,…) scale uncertainty Jäger,MS,Vogelsang all relevant observables available at NLO accuracy except for hadron-pair production at COMPASS, HERMES Q 2 ' 0 available very soon: Hendlmeier, MS, Schafer

8. 8 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: semi-inclusive DIS data so far only used in DNS fit ! flavor separation “classic” inclusive DIS data routinely used in PDF fits !  q +  q first RHIC pp data (never used before) ! g! g 467 data pts in total ( ¼ 10% from RHIC)

9. 9 data with observed hadrons SIDIS (HERMES, COMPASS, SMC) pp !  X (PHENIX) strongly rely on fragmentation functions Fortran codes of the DSS fragmentation fcts are available upon request ! new DSS FFs are a crucial input to the DSSV PDF fit  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]) 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:

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

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

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

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

14. 14 applicability & performance  computing load O(10 sec)/data pt. ! O(1 msec)/data pt. recall: need thousands of calls to perform a single fit ! production of grids much improved recently can be all done within a day with new MC sampling techniques  obtaining the grids once prior to the fit 64 £ 64 £ 4 £ 10 ' O(10 5 ) calls per pp data pt. nm n,m complex # subproc’s tested for pp !  X, pp !  X, pp ! jetX (much progress towards 2-jet production expected from STAR)  method completely general

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

16. 16 overall quality of the global fit  2 /d.o.f. ' 0.88 note: for the time being, stat. and syst. errors are added in quadrature very good! no significant tension among different data sets

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

18. 18 remark on higher twist corrections  we only account for the “kinematical mismatch” between A 1 and g 1 /F 1 in (relevant mainly for JLab data)  no need for additional higher twist corrections (like in Blumlein & Bottcher) at variance with results of LSS (Leader, Sidorov, Stamenov) – why?  very restrictive functional form in LSS:  f = N ¢ x  ¢ f MRST only 6 parameters for pdfs but 10 for HT  very limited Q 2 – range ! cannot really distinguish ln Q 2 from 1/Q 2  relevance of CLAS data “inflated” in LSS analysis: 633 data pts. in LSS vs. 20 data pts. in DSSV in a perfect world this should not matter, but …

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

20. 20 RHIC pp data (inclusive  0 or jet)  good agreement  important constraint on  g(x) despite large uncertainties ! later uncertainty bands estimated with Lagrange multipliers by enforcing other values for A LL

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

22. 22 Lagrange multiplier method see how fit deteriorates when PDFs are forced to give a different prediction for observable O i O i can be anything: we have looked at A LL, truncated 1 st moments, and selected fit parameters a j so far finds largest  O i allowed by the global data set and theoretical framework for a given  2 explores the full parameter space {a j } independent of approximations track  2  requires large series of minimizations (not an issue with fast Mellin technique)

23. 23  2 - a question of tolerance What value of  2 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  2 = 1 ! idealistic  2 =1 $ 1  approach usually fails we present uncertainties bands for both  2 = 1 and a more pragmatic 2% increase in  2 see: CTEQ, MRST, … also:  2 = 1 defines 1  uncertainty for single parameters  2 ' N par is the 1  uncertainty for all N par parameters to be simultaneously located in “  2 -hypercontour” used by AAC

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

25. 25 x a closer look at  u  small, mainly positive  negative at large x  2  determined by SIDIS data  pions consistent  mainly charged hadrons

26. 26 x a closer look at  s  positive at large x  negative at small x striking result!  2  determined by SIDIS data  mainly from kaons, a little bit from pions  DIS alone: more negative

27. 27 a closer look at  g error estimates more delicate: small-x behavior completely unconstrained x study uncertainties in 3 x-regions RHIC range 0.05 · x · 0.2 small-x 0.001 · x · 0.05 large-x x ¸ 0.2   g(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 find

28. 28 1 st moments: Q 2 = 10 GeV 2   s receives a large negative contribution at small x   g: huge uncertainties below x ' 0.01 ! 1 st moment still undetermined  SU(2)  SU(3)   SU(2),SU(3) come out close to zero

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

30. 30 Hessian method estimates uncertainties by exploring  2 near minimum: Hessian H ij taken at minimum displacement: only quadratic approximation  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 good news: can benefit from a lot of pioneering work by CTEQ and use their improved iterative algorithm to compute H ij J. Pumplin et al., PRD65(2001)014011

31. 31 PDF eigenvector basis sets S K § eigenvectors provide an optimized orthonormal basis to parametrize PDFs near the global minimum construct 2N par eigenvector basis sets S k § by displacing each z k by § 1 the “coordinates” are rescaled such that  2 =  k z k 2 cartoon by CTEQ sets S k § can be used to calculate uncertainties of observables O i

32. 32 comparison with uncertainties from Lagrange multipliers  tend to be a bit larger for Hessian, in particular for  g(x)  Hessian method goes crazy if asking for  2 >1  uncertainties of truncated moments for  2 =1 agree well except for  g

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

34. 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 +  0 data (plus  +,  -, …)  first di-jet data from STAR ! more precisely map  g(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. 35 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 2. further improving on uncertainties  Lagrange multipliers more reliable than Hessian with present data  Hessian method perhaps useful for  2 = 1 studies, beyond ??  include experimental error correlations if available

36. 36

37. 37 extra slides

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

39. 39 x meet the distributions:  d  fairly large  negative throughout  2  determined by SIDIS data  some tension between charged hadrons and pions

40. 40  2 profiles of eigenvector directions for a somewhat simplified DSSV fit with 19 parameters #1: largest eigenvector (steep direction in  2 ) … #19: smallest eigenvector (shallow direction in  2 ) significant deviations from assumed quadratic dependence

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

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