Nucleon structure from lattice QCD

Transcription

1 Nucleon structure from lattice QCD M. Göckeler, P. Hägler, R. Horsley, Y. Nakamura, D. Pleiter, P.E.L. Rakow, A. Schäfer, G. Schierholz, W. Schroers, H. Stüben, Th. Streuer, J.M. Zanotti QCDSF Collaboration apenext: Computational Challenges and First Physics Results, GGI, Firenze 08. February 2007 D. Pleiter (DESY) (QCDSF Collaboration) Nucleon Structure GGI / 35

2 Outline Introduction Axial Coupling Moments of Unpolarised Structure Functions Electro-magnetic Formfactors Transverse Spin Structure Retrospect and Outlook D. Pleiter (DESY) (QCDSF Collaboration) Nucleon Structure GGI / 35

3 Content Introduction Introduction Axial Coupling Moments of Unpolarised Structure Functions Electro-magnetic Formfactors Transverse Spin Structure Retrospect and Outlook D. Pleiter (DESY) (QCDSF Collaboration) Nucleon Structure GGI / 35

4 Introduction Introduction Lattice in principle allows for ab initio calculation of Moments of nucleon structure functions Nucleon form factors (electromagnetic FF, GFF) Good control of several systematic errors related to extrapolations needed: Infinite volume limit V Continuum limit a 0 Chiral limit m q 0 Calculations now feasible for 2-3 flavours of dynamical fermions thanks to Improved algorithms (e.g. mass preconditioning, multiple timescales) New capability computers (e.g. BG/L, apenext) This talk: selected results from QCDSF collaboration on nucleon structure D. Pleiter (DESY) (QCDSF Collaboration) Nucleon Structure GGI / 35

5 Introduction Simulation parameters Configurations with N f = 2 O(a)-improved dynamical quarks generated by UKQCD+QCDSF+DIK. m PS,sea = 340,..., 1170 MeV a = 0.07,..., 0.11 fm m PS,val = 340,..., 1240 MeV V = 1.4,..., 2.6 fm Simulations much closer to the physical quark mass Reasonably small lattice spacing Volumes at lower limit a / r m π 2 2 m K (a m PS ) 2 D. Pleiter (DESY) (QCDSF Collaboration) Nucleon Structure GGI / 35

6 Introduction Momenta and polarisations 3 initial state momentum: L 2π p = 3 choices for polarisations: 0 0 0, Γ = (1 + γ 4 )/2, Γ = (1 + γ 4 )iγ 5 γ 1 /2 Γ = (1 + γ 4 )iγ 5 γ 2 /2 17 different choices of momentum transfer q = p p Configurations at distance 5-10 trajectories, 16-4 different sources Use binning to eliminate auto-correlations D. Pleiter (DESY) (QCDSF Collaboration) Nucleon Structure GGI / 35

7 Introduction Calculation of matrix elements Matrix elements are determined from ratios: R(t, τ, p, p ) = C 3(t, τ, p [, p ) C2 (τ, p )C 2 (t, p ] 1/2 )C 2 (t τ, p ) C 2 (t, p ) C 2 (τ, p )C 2 (t, p )C 2 (t τ, p ) where C 2 (t, p ) = αβ Γ βα B α (t, p ) B β (0, p ) and C 3 (t, τ, p, p ) = αβ Γ βα B α (t, p )O(τ) B β (0, p ) Non-perturbative renormalisation using, e.g., Rome-Southampton method (RI -MOM scheme) D. Pleiter (DESY) (QCDSF Collaboration) Nucleon Structure GGI / 35

8 Introduction Example: v 2a vs. v 2b p=(1,0,0) p=(0,1,0) p=(0,0,0) p=(1,0,0) p=(0,1,0) 0 (u) v 2a (u) v 2b (u) v 2b t t t (β, κ sea) = (5.4, ), V= D. Pleiter (DESY) (QCDSF Collaboration) Nucleon Structure GGI / 35

9 Content Axial Coupling Introduction Axial Coupling Moments of Unpolarised Structure Functions Electro-magnetic Formfactors Transverse Spin Structure Retrospect and Outlook D. Pleiter (DESY) (QCDSF Collaboration) Nucleon Structure GGI / 35

10 Introduction Axial Coupling Form factor of the proton axial current: [ ] p, s A u d µ p, s = ū(p, s ) γ µ γ 5 G A (q 2 q µ ) + γ 5 G P (q 2 ) u(p, s) 2m N Axial coupling constant: g A = G A (0) Parton model interpretation: Fraction of the nucleon spin carried by the quarks q p, s A u d µ p, s = 2g A s µ = 2( u d)s µ Precise experimental value for g A D. Pleiter (DESY) (QCDSF Collaboration) Nucleon Structure GGI / 35

11 Axial Coupling Chiral effective field theory [QCDSF 2006] Evaluation of matrix element of isovector axial current within a low energy effective theory of QCD feasible: g A (m PS ) Calculation for finite spatial cubic box of length L: g A (m PS, L) = g A (m PS ) + g A (m PS, L) Lattice results not sufficient to fix all parameters phenomenological input required: Pion decay constant N mass splitting 0 and axial coupling c A Coupling B20 r (λ) D. Pleiter (DESY) (QCDSF Collaboration) Nucleon Structure GGI / 35

12 Axial Coupling Exploring finite size effects g A m =0.35 GeV m =0.14 GeV L [fm] Results for g A at β = 5.29, κ = m PS 0.6 GeV D. Pleiter (DESY) (QCDSF Collaboration) Nucleon Structure GGI / 35

13 Chiral extrapolation Axial Coupling L= L= 1.91 fm L= 1.27 fm L= 0.95 fm g A m 2 [GeV 2 ] Consistent description of lattice and experimental results Results for small quark masses (m 2 PS < 0.1 GeV2 ) and large volumes (L > 2 fm) needed D. Pleiter (DESY) (QCDSF Collaboration) Nucleon Structure GGI / 35

14 Content Moments of Unpolarised Structure Functions Introduction Axial Coupling Moments of Unpolarised Structure Functions Electro-magnetic Formfactors Transverse Spin Structure Retrospect and Outlook D. Pleiter (DESY) (QCDSF Collaboration) Nucleon Structure GGI / 35

15 Introduction Moments of Unpolarised Structure Functions 1 0 ( ) M dx x n 2 F u d (x, Q 2 ) = f EF;v S 2 n Q 2, gs (M) v S n (g S (M)) Matrix elements v n can be measured on the lattice [ ] N( p) O {µ 1 µ n} q Tr N( p) S := 2v (q)s n [p µ1 p µn Tr] Results have to be renormalised using a scheme (e.g. MS) and scale (e.g. 2 GeV). D. Pleiter (DESY) (QCDSF Collaboration) Nucleon Structure GGI / 35

16 Moments of Unpolarised Structure Functions Chiral extrapolation Fit function model taking pion cloud around nucleon into account: [Thomas et al., Detmold et al.] v RGI n (r 0 m PS ) = F vn (r 0 m PS ) F vn (x) = v RGI n ( 1 Cx 2 ln x 2 ) x 2 + (r 0 Λ χ ) 2 + a n x 2 D. Pleiter (DESY) (QCDSF Collaboration) Nucleon Structure GGI / 35

17 Moments of Unpolarised Structure Functions Lattice results for v RGI 2b Fit to β = 5.29 data, Λ χ = 500 MeV, experimental value fixed β=5.20 β=5.25 β=5.29 β= m PS 2 [GeV 2 ] Fit ansatz does not describe data very well No indication found for strong FSE D. Pleiter (DESY) (QCDSF Collaboration) Nucleon Structure GGI / 35

18 Content Electro-magnetic Formfactors Introduction Axial Coupling Moments of Unpolarised Structure Functions Electro-magnetic Formfactors Transverse Spin Structure Retrospect and Outlook D. Pleiter (DESY) (QCDSF Collaboration) Nucleon Structure GGI / 35

19 Introduction Electro-magnetic Formfactors Standard decomposition of nucleon electromagnetic matrix elements: [ p, s J µ p, s = u(p, s ) γ µ F 1 (q 2 ) + iσ µν q ] ν F 2 (q 2 ) u(p, s) 2M N We will consider, e.g. Proton form factors: 2 3 uγµ u 1 3 dγµ d Isovector form factors: Experimental interest: q 2 scaling Flavour dependence uγ µ u dγ µ d Disconnected terms cancel D. Pleiter (DESY) (QCDSF Collaboration) Nucleon Structure GGI / 35

20 Typical result 3 Electro-magnetic Formfactors dipole tripole 2 F 2 (v) Q 2 [GeV 2 ] Difference of fits small wrt to statistical errors D. Pleiter (DESY) (QCDSF Collaboration) Nucleon Structure GGI / 35

21 Electro-magnetic Formfactors Parametrisation of lattice results Fit data to F i (q 2 ) = A i (1 q 2 /M 2 i ) p Fit ansatz determines F (q) 1 (0), F (q) 2 (0) (q = u, d) Di-/tripole masses or, equivalently, the form factor radii Lattice data not precise enough to fix p Looking for minimal χ 2 suggests: p = 2 for F (u) 1 p = 3 for F (d) 1, F (u) 2, F (d) 2 D. Pleiter (DESY) (QCDSF Collaboration) Nucleon Structure GGI / 35

22 Electro-magnetic Formfactors Scaling of F (d) (u) 1 /F 1 / F 1 (u) F 1 (d) β=5.25, κ sea = β=5.29, κ sea = β=5.40, κ sea = Q 2 [GeV 2 ] Data suggests p = 2 and p = 3 to fit F (u) 1 and F (d) 1, respectively Flavour dependence also observed in fits to experimental data [Diehl et al., 2005] D. Pleiter (DESY) (QCDSF Collaboration) Nucleon Structure GGI / 35

23 Electro-magnetic Formfactors Scaling of µ (p) G (p) e (q 2 )/G (p) m (q 2 ) µ p G e p /Gm p Q 2 [GeV 2 ] Ignore disconnected terms Naive extrapolation (linear in quark mass) But: Naive extrapolation is not correct D. Pleiter (DESY) (QCDSF Collaboration) Nucleon Structure GGI / 35

24 Electro-magnetic Formfactors Dirac radius [r (v) 1 ] (r 1 (v) ) 2 [fm 2 ] m PS [GeV ] Calculation of isovector radii within a low energy effective theory of QCD predicts strong quark mass dependence D. Pleiter (DESY) (QCDSF Collaboration) Nucleon Structure GGI / 35

25 Electro-magnetic Formfactors Pauli radius [r (v) 2 ]2 Joined fit to [r (v) 2 ]2 and κ (v) : (r 2 (v) ) 2 [fm 2 ] m PS [GeV] D. Pleiter (DESY) (QCDSF Collaboration) Nucleon Structure GGI / 35

26 Electro-magnetic Formfactors Anomalous magnetic moment κ (v) 6 4 κ (v)norm m PS [GeV] D. Pleiter (DESY) (QCDSF Collaboration) Nucleon Structure GGI / 35

27 Electro-magnetic Formfactors F (p) 2 : Lattice vs. Experiment Experiment β=5.29, κ= β=5.29, κ= F 2 (p) Q 2 [GeV 2 ] Better agreement for m PS 300 MeV? D. Pleiter (DESY) (QCDSF Collaboration) Nucleon Structure GGI / 35

28 Content Transverse Spin Structure Introduction Axial Coupling Moments of Unpolarised Structure Functions Electro-magnetic Formfactors Transverse Spin Structure Retrospect and Outlook D. Pleiter (DESY) (QCDSF Collaboration) Nucleon Structure GGI / 35

29 Introduction Transverse Spin Structure Lattice methods allow for calculation of moments of the quark density ρ(x, b, s, S ) 1 ρ n (b, s, S ) = dx x n 1 ρ(x, b, s, S ) = 1 { ( 1 A n0 (b 2 2 ) + si Si A Tn0 (b 2 ) 1 ) 4m 2 b à Tn0 (b 2 ) + bj ɛji ) (S i m B n0 (b2 ) + si B Tn0(b 2 ) } + s i (2bi bj b2 δij )S j 1 m 2 à Tn0 (b2 ) momentum fraction x transverse spin s distance b from the center-of-momentum transverse spin S D. Pleiter (DESY) (QCDSF Collaboration) Nucleon Structure GGI / 35

30 Transverse Spin Structure Generalized form factor B T (n=1,2)0 (t). - / 0 1 ), + * ( ! " # $ % & ' D. Pleiter (DESY) (QCDSF Collaboration) Nucleon Structure GGI / 35

31 Transverse Spin Structure Quark densities in nucleon Parametrize lattice data using p-pole fits Fourier transformation to impact parameter b space 9 : * +, * +- * +. * ) * +. ) * +- = > = A B ) * +, ) * +, ) * +- ) * +. * * +. / * +- * +, ; <! " "! # $ % & ' ( G DE G NO F M K DE LJI F C A B C H D. Pleiter (DESY) (QCDSF Collaboration) Nucleon Structure GGI / 35

32 Content Retrospect and Outlook Introduction Axial Coupling Moments of Unpolarised Structure Functions Electro-magnetic Formfactors Transverse Spin Structure Retrospect and Outlook D. Pleiter (DESY) (QCDSF Collaboration) Nucleon Structure GGI / 35

33 Retrospect and Outlook APE and the QCDSF physics program APE machines at NIC/DESY (Zeuthen) APE APEmille apenext Selected bibliography: 1994: Towards a Lattice Calculation of the Nucleon Structure Functions 1997: Pion and Rho Structure Functions from Lattice QCD 1999: A Lattice Determination of Light Quark Masses 2000: A lattice calculation of the nucleon s spin-dependent structure function g2 revisited 2001: Determination of ΛMS from quenched and N f = 2 dynamical QCD 2002: A lattice study of the spin structure of the Lambda hyperon 2003: Nucleon electromagnetic form factors on the lattice and in chiral effective field theory 2004: A lattice determination of moments of unpolarised nucleon structure functions using improved Wilson fermions 2005: Quark helicity flip generalized parton distributions from two-flavor lattice QCD 2006: Moments of pseudoscalar meson distribution amplitudes from the lattice QCDSF program was only possible with APE D. Pleiter (DESY) (QCDSF Collaboration) Nucleon Structure GGI / 35

34 Retrospect and Outlook International Lattice Datagrid Huge efforts required to push towards m q 0, V, a 0 Collaborations start to become larger Different sources for compute power used Important to improve on data sharing A number of relevant standards have been defined within ILDG Started to build-up a (continental) European infrastructure which starts to become heavily used: SESAM/TχL/GRAL ETMC QCDSF total O(50,000) O(25,000) O(20,000) O(95,000) MWWG goal: Interoperability for download operations by LAT 07 D. Pleiter (DESY) (QCDSF Collaboration) Nucleon Structure GGI / 35

35 Retrospect and Outlook Conclusions and outlook Lattice continues to be interesting method to explore hadron structure: Axial coupling g A, moments of unpolarised structure functions Electro-magnetic form factors and GFF (transverse spin structure) Hadron structure is hot topic for various experimental programs (e.g. JLAB, GSI) Control on systematic errors improved but is often not sufficient Possibly large finite size effects Strong Quark mass dependence for mps m π chiral extrapolations critical Discretisation effects seem to be relatively small O(a) improvement program successful Exploring the light quark mass region starts to become feasible Exciting prospect But: Computing power O(100) TFlops (sustained) required D. Pleiter (DESY) (QCDSF Collaboration) Nucleon Structure GGI / 35