1. Parametrization of Generalized Parton Distributions Simonetta Liuti* University of Virginia ECT* Workshop on GPDs 11 th -15 th October 2010 *Work in collaboration with Gary Goldstein, Osvaldo Gonzalez Hernandez, Kathy Holcomb, Kunal Kathuria

2. From toy models  to quantitative extractions Summary of “1 st generation parametrizations” Diehl, Feldmann, Jacob, Kroll (2004); Ahmad, Honkanen, Liuti, Taneja (2006,2009); Guidal, Polyakov, Radyushkin, Vanderhaeghen (2005)  New Extraction Methods: Global Analyses H. Moutarde’s and collaborators Approach D. Mueller’s and collaborators Approach Recursive Fitting Procedure New! Neural Networks and Self-Organizing Maps G. Goldstein and S.L., Phys. Rev.D (2009), G. Goldstein and S.L., arXiv:1006.0213 (2010) Outline New!

3. Preamble q k' + =(X- Δ)P +, k T - Δ T q'=q+Δ k + =XP +, k T P' + =(1- Δ )P +, - Δ T P+P+

4. Amplitude Off forward Parton Distributions (GPDs) are embedded in soft matrix elements for deeply virtual Compton scattering (DVCS) p + =XP + p’ + =(X- Δ )P + P’ + =(1- Δ )P + P+P+ p+q q q’=q+ Δ

5. Re k - Im k - 3'1' 2' Re k - Im k - 2'3' 1' X>ζ X<ζ 3 1 2

6. P+P+ k’ + =(  -X)P + P’ + =(1-  )P + In ERBL region struck quark, k, is on-shell k + =XP + P+P+ k’ + =(X-  )P + P’ + =(1-  )P + P X + =(1-X)P + In DGLAP region spectator with diquark q. numbers is on-shell Analysis done for DIS/forward case by Jaffe NPB(1983)

7. P+P+ k’ + =(  -X)P + P’ + =(1-  )P + ERBL region corresponds to semi-disconnected diagrams: no partonic interpretation

8. GPDs in ERBL region can be described within QCD, consistently with factorization theorems, only by multiparton configurations

9. Dispersion relations cannot be applied straighforwardly to DVCS. The “ridge” does not seem to contain all the information Gary Goldstein’s talk

10. A bit of history…summary of previous results Guidal, Polyakov, Radyushkin, Vanderhaeghen PRD (2005) Diehl, Feldmann, Jacob, Kroll EPJC (2004) Combined x and t dependence From Gaussian shape of w.f.’s in k T : consistent with Regge behavior at small x! 2005/06 First non-trivial (non factorized x and t dependences) parametrizations appear that fit form factors and predict angular momentum, J u and J d. PDF only ζ =0!

11. 1/2

12. Ahmad, Honkanen, S.L., Taneja PRD(2006) 2006, I n order to go to ζ≠0 one needs to model the amplitude “from scratch”: covariant quark-diquark model k + =XP + k’ + =(X-ζ)P + P’ + =(1- ζ)P + P X + =(1-X)P + P+P+ S=0 or 1 ERBL region DGLAP region

13. Predict J u and J d

14. First “shapes” of GPDs

15. ERBL Region Ahmad et al., EPJC (2009) Determined from lattice moments up to n=3

16. In doing so … a large set of increasingly complicated and diverse observations is involved, from which PDFs, TMDs, GPDs, etc… are extracted and compared to patterns predicted theoretically What goes into a theoretically motivated parametrization for GPDs...? The name of the game: Devise an analytical form combining essential dynamical elements with a flexible model whose parameters can be tuned However…for a fully quantitative analysis that is constrained by the data it is fundamental to evaluate both the number and type of parameters Multi-dimensional phase space (Elliott Leader’s talk)

17. “Conventional” based on microscopic properties of the theory (two body interactions) “Neural Networks” study the behavior of multiparticle systems as they evolve from a large and varied number of initial conditions. Two main approaches “Regge” Quark-Diquark + Q 2 Evolution Proceed conventionally at first: Ahmad et al.,

18. Covariant quark diquark model Re k - Im k - 2'3' 1' X>ζ 3 1 2

19. Skip some neat details, derivation from helicity amps. and connection with Chiral Odd GPDs (Kubarovsky’s pi0 data)

20. Reggeization DIS Brodsky, Close, Gunion ‘70s Diquark spectral function ρ MX2MX2 (M X 2 ) α δ(M X 2 -M X 2 )

21. Crossing Symmetries

22.  /2  H-H- H+H+ (H + + H - )/2=H q Q 2 dependent dominated by valence

23. Recursive Fitting Procedure vs. Global Fits In global fits we apply constraints simultaneously  calculate χ 2 (and covariance matrix) once including all constraints  Need to find the solution for a system with n equations Everytime new data come in the fit needs to be repeated from scratch  In a recursive fitting procedure (Kalman) one applies constraints sequentially  Need to update solution after each constraint  Find solutions for < n systems, each one with fewer equations Better control at each stage Mandatory for GPDs (some of this is implicit in Moutarde’s approach)

24. 1 st stepConstraints from PDFs Note! This is accomplished with 5 parameters per quark flavor, per PDF

25. LSS06

26. 2 nd stepConstraint from FFs α’ = α’ o (1-X) p

27. Constraints from Crossing Symmetries H

28. 7 parameters per quark flavor per GPD, but we have not constrained the ζ dependence yet….. DVCS data

29. Observables

30. σ, A C, A LU A LU A UT σ, A C, A LU

32. Hall B and Hall A A LU Extract “a” from data

33. Hall A Hall B Our (GGL) prediction based on our choice of physical model, fixing parameters from “non-DVCS” data

34. H+E E

35. Neural Networks (NN) have been widely applied for the analysis of HEP data and PDF parametrizations When applied to data modeling, NNs are a non-linear statistical tool The network makes changes to its connections upon being informed of the “correct” result via a cost/object function. Cost function  measures the importance to detect or miss a particular occurrence Example: If all patterns have equal probability, then the cost of predicting pattern S i instead of S k is simply In general the aim is to minimize the cost  learning process Most NNs (including NNPDFs) learn with supervised learning The Use of Neural Networks in Data Analysis

36. Unsupervised Learning No a priori examples are given. The goal is to minimize the cost function by similarity relations, or by finding how the data cluster or self-organize  global optimization problem Important for PDF analysis! If data are missing it is not possible to determine the output! We introduced a type of NN  the Self-Organizing Map that is based on:

37. ViVi

38. SOMPDFs SOMPDF.0 J. Carnahan, H. Honkanen, S.L., Y. Loitiere, P. Reynolds, Phys Rev D79, 034022 (2009) SOMPDF.1, K. Holcomb, S.L., D.Z.Perry, hep-ph (2010)

39. Uncertainties from different PDF evaluations/extractions (Δ PDF ) are smaller than the differences between the evaluations (Δ G ) Δ PDF < Δ G d-bar u-valenced-valence Gluon Main issue

40. Similarly to NNPDFs we eliminate the bias due to the initial parametric form

42. First Results (on arxiv in 8/2010) (D. Perry, DIS 2010 and MS Thesis 2010, K. Holcomb, Exclusive Processes Workshop, Jlab 2010) s u bar uvuv dvdv Q 2 = 7.5 GeV 2

43. Comparison with MSTW08

44. Extension to GPDs SOMs differently from standard ANN methods are “unsupervised”  they find similarities in the input data without a training target. They have been used in theoretical physics approaches to critical phenomena, to the study of complex networks, and in general for the study of high dimensional non-linear data (see e.g. Der, Hermann, Phys.Rev.E (1994), Guimera et al., Phys. Rev.E (2003) ) “Study of particle shape and size” N. Laitinen et al., 2001

45. We are studying similar characteristics of SOMs to devise a fitting procedure for GPDs: our new code has been made flexible for this use Main question: Which experiments, observables, and with what precision are they relevant for which GPD components? From Guidal and Moutarde, and Moutarde analyses (2009) 17 obsvervables (6 LO) from HERMES + Jlab data 8 GPD-related functions “a challenge for phenomenology…” (Moutarde) + “theoretical bias”

46. Preliminary: the 8 GPDs are the dimensions in our analysis E Im H˜ I m E˜ Im E Re H Im H˜ Re E˜ Re H Re

47. Conclusions Interesting new aspects of partonic interpretation of hadrons (agree with Oleg’s initial statements!) Fully quantitative analyses ready! Applications to chiral odd (Kubarovsky, Nowak) on their way! Developed SOMPDFs SOMGPDs on their way!