Nucleon Structure from Lattice QCD

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1 Nucleon Structure from Lattice QCD Philipp Hägler supported by excellence cluster universe

2 Charges, magnetic moments, (r)ms radii,... Form factors PDFs Distribution in (longitudinal) momentum u xp z q/g spin content/distribution u P z long. momentum fraction d z GPDs Distribution of momentum&spin In coordinate space b y u xp z Orbital angular momentum Ph. Hägler, MENU 2010, W&M 2 b x b d u z P z

3 Charges, magnetic moments, (r)ms radii,... Form factors Wigner phase-space-distributions [X. Ji, PRL 2003; A. Belitsky, X. Ji, F. Yuan, PRD 2004 ] mother-distributions [Meissner, Metz, Schlegel JHEP 0908:056,2009] PDFs Distribution in (longitudinal) momentum u xp z q/g spin content/distribution u P z long. momentum fraction d z TMDs GPDs Distribution of momentum&spin In Distribution coordinate of space momentum&spin in transverse momentum space Orbital angular momentum T-odd effects b y b x Ph. Hägler, MENU 2010, W&M 3 u b u d xp z z P z

4 QCD factorization and observables on the lattice DIS bilocal operators on the light-cone on a Euclidean space-time lattice SIDIS first exploratory studies of non-local couplings on the lattice related to TMDs [Musch(Mo 1E 2pm), PhH, Schäfer, Negele] x-moments matrix elements of local (quark) operators DVCS (generalized) form factors x-moments of PDFs, GPDs measurements at HERMES/DESY, COMPASS/CERN, Jefferson Lab, RHIC Ph. Hägler, MENU 2010, W&M 4

5 Measurements on the lattice Ph. Hägler, MENU 2010, W&M 5 gauge fields/links U quarks quark propagators irrel. for u-d [Doi (Wed 6A 4:10pm)] products of quark propagators compute the path-integral numerically using MC methods

6 Lattice QCD general comments Ph. Hägler, MENU 2010, W&M 6 systematic ab initio -approach, but statistical errors from MC integration discretization and finite volume errors/effects contaminations from excited states unphysical quark masses large minimal non-zero momenta [Aubin (Wed 17:10pm)] [W.-Loud (Wed 17:30pm)] [Edwards (Fr 8:30am)] Lattice QCD is different from model calculations approximations can be continuously improved mainly limited by computational and human resources

7 some human resources I am indebted to: Ph. Hägler, MENU 2010, W&M 7 M. Altenbuchinger, B. Musch ( JLab), M. Gürtler ( Regensburg), W. Weise (T39, TUM) B. Bistrovic, J. Bratt, J.W. Negele, A. Pochinsky, S. Syritsyn (MIT) R.G. Edwards, B. Musch, D.G. Richards (JLab) K. Orginos (W&M) M. Engelhardt (New Mexico) G. Fleming, M. Lin (Yale), H.-W. Lin (INT), H. Meyer (Mainz), D.B. Renner (DESY Zeuthen), M. Procura (TUM), W. Schroers (LHPC) D. Brömmel (Southampton), M. Diehl (DESY), M. Göckeler, M. Gürtler, Th. Hemmert, A. Schäfer (Regensburg U.) R. Horsley, J. Zanotti (Edinburgh U.) Y. Nakamura (DESY Zeuthen) P. Rakow (Liverpool U.) D. Pleiter, G. Schierholz (DESY) H. Stüben (ZIB) (QCDSF/UKQCD) References: QCDSF PoS(LAT2006)120, , PRL (2007), PRL 2008 ( ), Brömmel et al EPJC 2007; LHPC PRD 77, (2008), ; ; Diehl&Hägler EPJC hep-ph/ ; Musch et al ; Musch arxiv: ; PhH, Musch et al. EPL 2009 (arxiv: ) PhH Phys.Rep ( )

8 Machines SGI Altix 4700 at LRZ Garching APEmille at NIC/DESY Zeuthen 7n cluster at JLab Blue Gene (MIT, EPCC, ) Ph. Hägler, MENU 2010, W&M sources: ( & Jlab, EPCC webpages 8

9 Overview Ph. Hägler, MENU 2010, W&M 9 form factors nucleon rms radii and (anomalous) magnetic moment axial vector and tensor coupling constants GPDs generalized form factors and the form factors of the energy momentum tensor decomposition of the nucleon spin correlations between momenta, positions and spins PhH Phys.Rep ( )

10 Proton mean square radii Dirac isovector radius Dirac and Pauli FFs r 2 i 6 F (0) i d dt F ( t) i t0 SSE ChPT Hemmert, Weise EPJC 02 Göckeler et al PRD 93 preliminary QCDSF 2010 (J. Zanotti et al.) 1-loop HBChPT Bernard, Kaiser, Meissner NPB 1993 [Syritsyn(Mo 1E 3pm) for LHPC] Ph. Hägler, MENU 2010, W&M 10

11 Nucleon isovector anomalous magnetic moment Ph. Hägler, MENU 2010, W&M 11 (anomalous) magnetic moment exp included in fit SSE ChPT Hemmert, Weise EPJC 02 Göckeler et al PRD 93 preliminary QCDSF 2010 (J. Zanotti et al.)

12 Proton mean square radii Pauli isovector radius Dirac and Pauli FFs SSE ChPT Hemmert, Weise EPJC 02 Göckeler et al PRD 93 (using ) preliminary QCDSF 2010 (J. Zanotti et al.) [Young (Fr 9am)] Ph. Hägler, MENU 2010, W&M 12

13 Nucleon axial vector coupling constant n f 2 1 DW RBC UKQC n f 2 1 DW LHPC '08 n f 2 1 mixed LHPC '08 n f 2 1 mixed LHPC '06 n f 2 DW RBC n f 2 Clover QCDSF '10 preliminary exp. g A -landscape detailed ChPT analysis necessary Ph. Hägler, MENU 2010, W&M 13

14 Tensor charge preliminary QCDSF 2010 (J. Zanotti et al.) see also recent ral QSM calc. Ledwig et al. ( ) [Prokudin(Mo 1E 2:20pm)] Ph. Hägler, MENU 2010, W&M 14

15 Higher (x n-1 -) moments of GPDs Ph. Hägler, MENU 2010, W&M 15 GFFs Ji&Lebed PRD 2000 Ph.H. PLB 2004 form factors of the EM-tensor Decomposition of the nucleon spin Ji s nucleon spin sum rule (X. Ji, PRL 1997) everything is: -gauge-invariant -scale & scheme dependent -measurable

16 Ph. Hägler, MENU 2010, W&M 16 b y u b Generalized mean square radii of the nucleon correlations in x and b from n=1,2,3 xp z b x d u z P z LHPC n f =2+1 mixed; arxiv: (updating PRD 2008, and PRL 93, hep-lat/ ) strong correlations in x and b no factorization of GPDs in x and t

17 Ph. Hägler, MENU 2010, W&M 17 n=2 - A, B, C - Form factors of the energy momentum tensor LHPC n f =2+1 mixed; arxiv: (updating PRD 2008, ) disconnected contributions are not included only u-d is exact seems to be compatible with large N c limit see e.g. Goeke, Polyakov, Vanderhaeghen PiPaNP 2001

18 Quark angular momentum Ph. Hägler, MENU 2010, W&M 18 from covariant BChPT extrapolations Dorati, Gail, Hemmert NPA 2008 LHPC arxiv: preliminary LHPC DW (Syritsyn et al.) LHPC n f =2+1 mixed; arxiv: (updating PRD 2008, )

19 Ph. Hägler, MENU 2010, W&M 19 J u, J d template figure [JLab Hall A PRL`07; HERMES JHEP`08] LHPC arxiv: (this work) LHPC PRD ` QCDSF (Ohtani et al.) Goloskokov&Kroll EPJC` Wakamatsu DiFeJaKr EPJC `05 hep-ph/ (Myhrer&)Thomas PRL`

20 Quark spin and orbital angular momentum LHPC n f =2+1 mixed arxiv: pioneering lattice calculations by Gadiyak, Ji and Jung in 2001 Ph. Hägler, MENU 2010, W&M 20

21 Correlations between momenta, positions, spins Ph. Hägler, MENU 2010, W&M 21 center of momentum generalized parton distributions (GPDs) Fourier-transformation Burkardt PRD 2000 transverse momentum dependent PDFs

22 b y u b xp z Transverse spin densities in the proton in impact parameter space b x d u P z z Diehl / PhH EPJC 2005 lattice calculations of quark spin-flip couplings quark transverse spin in (+x)-direction QCDSF (PhH et al.) n f =2 Clover; PRL fm Femto-photography of quarks in the proton (Pire&Ralston PRD 2002) up down strongly deformed perpendicular to transverse spin Ph. Hägler, MENU 2010, W&M 22

23 Intrinsic transverse momentum densities of the nucleon Ph. Hägler, MENU 2010, W&M 23 first exploratory lattice study of transverse momentum distributions of quarks in the nucleon using non-local operators not identical to TMDs in exp. k y u k xp z Diehl, PhH EPJC 44 (2005) k x u d z P z up down Boglione, Mulders PRD 60 (1999) PhH, B. Musch et al. EPL; arxiv: visibly deformed parallel to transverse spin [Musch(Mo 1E 2pm) GPDs genuine effect of intrinsic transverse momentum of quarks

24 Summary remarkable progress in lattice QCD calculations of hadron structure? Dirac and Pauli radii tension with (H)BChPT?... or = freefoto.com large values for tensor charge tension with TMD-phenomenology * strong correlations between coordinate, momentum & spin DOFs Challenges evidence for substantial genuine intrinsic k T,... systematic uncertainties in benchmark observables (radii, magnetic moment, g A,...) disconnected diagrams; gluon operators; operator mixing * [non-singlet, connected only; additional uncertainties due to chiral extrapolations, renormalization] Ph. Hägler, MENU 2010, W&M under investigation (Syritsyn et al.) 24

25 Backup Ph. Hägler, MENU 2010, W&M 25

26 Decomposition of the proton spin * * [non-singlet, connected only; additional uncertainties due to chiral extrapolations, renormalization] under investigation (Syritsyn et al.) Ph. Hägler, MENU 2010, W&M 26

27 Lattice QCD vs relativistic quark models QCD evolution (Wakamatsu 2005; Thomas, PRL 2008) Ph. Hägler, MENU 2010, W&M 27

28 Lattice QCD vs relativistic quark models QCD evolution (Wakamatsu 2005; Thomas, PRL 2008) L u d J g 2 in coll. with LHPC and M. Altenbuchinger, W. Weise (TUM) lattice + evolution relativistic quark model Ph. Hägler, MENU 2010, W&M 28

29 Lattice simulation details Ph. Hägler, MENU 2010, W&M 29 - mixed action approach: DW fermions on a Asqtad staggered sea for N f =2+1; including HYP-smearing - L s =16, m res 0.1m q - lattice spacing a ~ fm - volumes of ~2.5 and ~3.5 fm 3 - two sink momenta P =(0,0,0), (-1,0,0) operator renormalization: # of measurements increased by factor 8 compared to PRD (2008) ongoing efforts within LHPC based on DW fermions (RBC/UKQCD) and improved Wilson fermions (BMW)

30 Ph. Hägler, MENU 2010, W&M 30 FRR

31 Gluon contributions to the proton spin * Burkardt et al. RPP 73:016201,2010 DSSV PRL 101:072001,2008; PRD 80:034030,2009 (?) * [non-singlet, connected only; additional uncertainties due to chiral extrapolations, renormalization] Ph. Hägler, MENU 2010, W&M 31

32 Form factors of the energy momentum tensor isovector quark momentum fraction Ph. Hägler, MENU 2010, W&M 32 compared to LHPC PRD (2008) LHPC arxiv: HBChPT-fit

33 Isosinglet quark spin fraction (required for L=J-Σ/2) Ph. Hägler, MENU 2010, W&M 33 employing HBChPT by Diehl, Manashov, Schäfer EJPA 2006; Ando, Chen, Kao PRD 2006 LHPC arxiv: HERMES PRD 2007

34 Quark angular momentum Ph. Hägler, MENU 2010, W&M 34 emplyoing HBChPT+ results [Chen Ji PRL 2002] LHPC arxiv:

35 Isovector axial vector coupling constant (required for L=J-Σ/2) Ph. Hägler, MENU 2010, W&M 35 emplyoing SSE (HBChPT+) results [Procura, Hemmert, Musch, Weise PRD 2007, QCDSF PRD 2006] LHPC arxiv: compare to LHPC PRL (2006)

36 Quark spin and OAM Ph. Hägler, MENU 2010, W&M 36 LHPC PRD D 77, (2008) preliminary LHPC DW (Syritsyn et al.) relativistic quark models LHPC n f =2+1 LHPC mixed PRD 2003 arxiv: (updating PRD 2008)

37 Global, simultaneous chiral extrapolation of A, B, C Ph. Hägler, MENU 2010, W&M 37 LHPC n f =2+1 mixed; arxiv: (updating PRD 2008, ) only quark line connected contributions chiral extrapolation based on covariant BChPT by Dorati, Gail, Hemmert NPA 2008 with common parameter

38 Nucleon axial vector coupling constant Ph. Hägler, MENU 2010, W&M 38 J. Zanotti et al. [QCDSF], PoS Lat2009

39 B 20 and the anomalous gravitomagnetic moment based on HBChPT by Diehl, Manashov, Schäfer EJPA 2006 Ando, Chen, Kao PRD 2006 [Teryaev `99-; Brodsky, Hwang et al. `00-] including LHPC n f =2+1 mixed arxiv: (updating PRD 2008) O(p 3 ) non-linear correlation in t and m small quark AGM Ph. Hägler, MENU 2010, W&M 39!?

40 C 20 and the second moment of the D-term [Polyakov&Weiss `99] based on HBChPT by Diehl, Manashov, Schäfer EJPA 2006 Ando, Chen, Kao PRD 2006 including LHPC n f =2+1 mixed arxiv: (updating PRD 2008) O(p 3 ) non-linear correlation in t and m sizeable and negative Ph. Hägler, MENU 2010, W&M 40!?

41 Ph. Hägler, MENU 2010, W&M 41 Nucleon axial vector coupling constant published data g A -landscape n f 2 1 DW RBC UKQCD n f 2 1 DW LHPC '08 n f 2 1 mixed LHPC '08 n f 2 1 mixed LHPC '06 n f 2 DW RBC UKQCD n f 2 Clover QCDSF '06 exp.

42 Transversely polarized quarks in transversely polarized nucleons Ph. Hägler, MENU 2010, W&M 42 probability density for (transversely polarized) quarks in a (transversely polarized) proton multipole-expansion dipole-term Diehl / Burkardt PhH EPJC monopole dipole quadrupole

43 Intrinsic transverse momentum densities of the nucleon Diehl, PhH EPJC 44 (2005) Boglione, Mulders PRD 60 (1999) Ph. Hägler, MENU 2010, W&M 43

44 correlationsin x and t Ph. Hägler, MENU 2010, W&M 44 x 1 n compared to parametrization/ansatz by Diehl, Feldmann, Jakob, Kroll EPJC 2005 A A ud 30 ud 10 ( t) ( t)

45 Ph. Hägler, MENU 2010, W&M 45 Nucleon form factors ) ( ) ( 2 ) ( ) ' ( ' 2 1 P U t F m i t F P U q P q P isovector ˆ,, p d d u p u q n q e n p q q e p u d q q u d q q remember Dirac and Pauli FFs ˆ ) ( ) ( ) ( ) ( 4 ) ( ) ( Q t t F t F t G t F m t t F t G M N E where electric and magnetic FFs ) 0 ( 0) ( 1 t F t G M (anomalous) magnetic moment charge radii ) ( (0) 6 t t F dt d F r n p N n E p E d u p m r r r e.g. p M p E G G, n M n E G G, d u M d u G E G, b u d P z z u b x b y probability densities

46 Lattice QCD propaganda Ph. Hägler, MENU 2010, W&M 46 BMW (Dürr et al.) Nature 2009

47 Lattice QCD propaganda Ph. Hägler, MENU 2010, W&M 47 Davies, Lepage et al. PRL 2004 Davies, Lepage et al. PRL 2008

48 Lattice parameters LHPC - domain - wall - fermions ("hybrid"formalism) with HYP - smearing - use of staggered quarks is "a matter of taste" - N f 2 1, but only connected contributions - Ls 16, m res 01. m -inverse lattice -spacing is a ~ 1 - one projector pol ( 1 0)( 1 5 3) 4 - twosink - momenta p' ( 0, 0, 0),( 10,, 0) q on a staggered 1.6 GeV "Asqtad"staggered - pion masses as low as 300M ev in volumes (3.5fm) 1 3 sea operator renormalization: # of measurements increased by factor 8 compared to PRD (2008) x / (7) 292 Ph. Hägler, MENU 2010, W&M 48

49 Form factors of the energy momentum tensor and fundamental sumrules graviton-coupling spin-2 coupling translation invariance rotational symmetry conservation of momentum conservation of angular momentum momentum sumrule vanishing of the anomalous gravitomagnetic moment Ji s nucleon spin sum rule everything is: -gauge-invariant -scale and scheme dependent -measurable Ph. Hägler, MENU 2010, W&M 49

50 Form factors of the energy momentum tensor isovector quark momentum fraction Ph. Hägler, MENU 2010, W&M 50 compared to LHPC PRD (2008) HBChPT-fit

51 Ph. Hägler, MENU 2010, W&M 51 TMDs in lattice QCD PhH, B. Musch, J. Negele, A. Schäfer, arxiv: B. Musch, PhD thesis arxiv:

52 Transverse momentum dependent PDFs - formalism Ph. Hägler, MENU 2010, W&M 52 k y k u xp z k x u d z P z complex amplitude A 2 choice of Wilson line U semi-inclusive DIS FSI lattice (so far) direct link

53 Transverse momentum dependent PDFs - formalism Ph. Hägler, MENU 2010, W&M 53 Mulders, Tangermann NPB 1996 Boer, Mulders PRD 57 (1998) Fourier-transformation

54 Overview of numerical results for A 2 all for m ~ 500 MeV LHPC, N f =2+1 DW-valence +staggerd sea Ph. Hägler, MENU 2010, W&M 54

55 Renormalization Ph. Hägler, MENU 2010, W&M 55 potential power-divergence renormalization condition

56 Illustration of renormalization Ph. Hägler, MENU 2010, W&M 56 renormalized

57 Regularization and multiplicative renormalization Ph. Hägler, MENU 2010, W&M 57 Gaussian parametrization of the invariant amplitudes restricted to l ~ fm reg. potential divergences at small l large k renormalize multiplicatively such that at this stage, better not compare quantitatively with TMD-phenomenology (e.g. Anselmino et al.)

58 l 2 -dependence of invariant amplitudes (renormalized) PhH, B. Musch et al. arxiv: comp. to LC quark model: Pasquini et al PRD 2008 Ph. Hägler, MENU 2010, W&M 58

59 Intrinsic transverse momentum densities of the nucleon Ph. Hägler, MENU 2010, W&M 59 Musch et al. n f =2+1 mixed preliminary and PoSLC2008 GPDs genuine effect of intrinsic transverse momentum of quarks

60 Approximate relations between GPDs and TMDs Ph. Hägler, MENU 2010, W&M 60 T-odd Sivers T-odd Boer-Mulders Burkardt PRD 2002 Diehl, PhH EPJC 2005 Metz et al. 2007

61 Momentum fraction of quarks in the nucleon Ph. Hägler, MENU 2010, W&M 61 substantial systematic uncertainties

62 Intrinsic transverse momentum densities of the nucleon Ph. Hägler, MENU 2010, W&M 62 up down PhH, B. Musch et al. arxiv: GPDs genuine effect of intrinsic transverse momentum of quarks

63 Transverse momentum dependent PDFs correlations in x and k Musch et al. n f =2+1 mixed tbp and PoS LC2008 Ph. Hägler, MENU 2010, W&M 63 no visible correlations in l P and l 2 factorization of tmdpdfs in x and k

64 Spin structure of the pion pion spin sumrule QCDSF n f =2 Clover PRL 2008 Is the pion spin structure trivial? but is non-zero? quark transverse spin in x-direction b y u b xp z P z b x d z up-quarks in a + QCDSF n f =2 Clover, PRL 2008 lattice calculations of quark spin-flip couplings the pion has a non-trivial transverse spin structure! Ph. Hägler, MENU 2010, W&M 64