nucleon parton distributions from lattice qcd

Nucleon parton distributionsfrom lattice QCD

Philipp Hägler

supported by

Ph. Hägler, IWHSS, Paris 04/11

2

Motivation

• get deeper insight into inner quark and gluon structure of hadrons

• learn about fundamental aspects and mechanisms of the strong interaction

• facilitate understanding and interpretation of phenomenology of hadron scattering and production; comparison with experiment

• make predictions; support search for „new physics“

• provide an approach that is complementary/alternative to the „triumvirate“ pQCD + QCD factorization + experiment as well as other NP-approaches

• validate and improve lattice methods and techniques

• ...

• get deeper insight into inner quark and gluon structure of hadrons

• learn about fundamental aspects and mechanisms of the strong interaction

• facilitate understanding and interpretation of phenomenology of hadron scattering and production; comparison with experiment

• make predictions; support search for „new physics“

• provide an approach that is complementary/alternative to the „triumvirate“ pQCD + QCD factorization + experiment as well as other NP-approaches

• validate and improve lattice methods and techniques

• ...


3

DIS SIDIS DVCS

QCD factorization and observables on the lattice

bilocal operators on the light-conebilocal operators on the light-cone on a Euclidean space-time latticeon a Euclidean space-time lattice

x-momentsx-moments

matrix elements of local (quark) operatorsmatrix elements of local (quark) operators

(generalized) form factors(generalized) form factors

x-moments ofPDFs, GPDs

x-moments ofPDFs, GPDs

use manifestly non-local operatorsoff the light-cone on the lattice

use manifestly non-local operatorsoff the light-cone on the lattice

approach light-cone limit usingextrapolations/theory

approach light-cone limit usingextrapolations/theory

invariant amplitudesinvariant amplitudes

TMDsTMDs(?)


4

Lattice QCD: Introduction

hadron properties&structurehadron properties&structure

hadron masseshadron masses

form factorsform factors

moments of (generalized) PDFsmoments of (generalized) PDFs

decay constantsdecay constants

polarizabilitiespolarizabilities

moments of distribution amplitudesmoments of distribution amplitudes

k-dependent distributions (TMDs)k-dependent distributions (TMDs)pion, nucleon, , pion, nucleon, ,

hadronshadrons

Nf=2 and 2+1, many differentlattice actions (Wilson, DW,...)

with m≈300MeV standard

Nf=2 and 2+1, many differentlattice actions (Wilson, DW,...)

with m≈300MeV standard

lattice computationslattice computations

systematic„ab initio“-approach; still with:

systematic„ab initio“-approach; still with:

discretization errors/effects finite volume errors/effects large quark masses ...

discretization errors/effects finite volume errors/effects large quark masses ...

Euclidean

significant progress in hadron structure calculations on the lattice during last 10 yearssignificant progress in hadron structure calculations on the lattice during last 10 years


5

gauge fields/links U

quark propagators

quarks

= vector-, axialvector-, quark spin flip-, (spin-2) graviton-, „spin-n“ coupling

Kentucky (Liu et al.), Collins Bali Schäfer,...Kentucky (Liu et al.), Collins Bali Schäfer,...

Typical „measurements“ on the lattice

compute the path-integral using MC methodscompute the path-integral using MC methods


6

Not quite as trivial as it might appear...


7

Past and ongoing lattice studies

pion and nucleon form factorspion and nucleon form factors

QCDSF/UKQCDRBC/UKQCD

LHPCMIT/Cyprus

ETMCJLQCD

CLS (Mainz)...

QCDSF/UKQCDRBC/UKQCD

LHPCMIT/Cyprus

ETMCJLQCD

CLS (Mainz)...

moments of PDFsmoments of PDFs

QCDSF/UKQCDLHPC

RBC/UKQCDETMC

...

QCDSF/UKQCDLHPC

RBC/UKQCDETMC

...

moments of GPDsmoments of GPDsQCDSF/UKQCD

LHPC(ETMC?)

...

QCDSF/UKQCDLHPC

(ETMC?)...

reviews: Zanotti 0812.3845; Renner 1002.0925PhH Phys.Rept. 2010 (0912.5483)

reviews: Zanotti 0812.3845; Renner 1002.0925PhH Phys.Rept. 2010 (0912.5483)concentrate on small number of selected resultsconcentrate on small number of selected results

„TMDs on the lattice“„TMDs on the lattice“ PhH, Musch, Negele, SchäferPhH, Musch, Negele, Schäfer


8

Isovector momentum fraction

substantial systematic uncertainties: renormalization, discretization errors, excited state contamination, ...?substantial systematic uncertainties: renormalization, discretization errors, excited state contamination, ...?


9

Form factors of the energy momentum tensor

QCDSF/UKQCD nf=2A.Sternbeck, D.Pleiter, J.Zanotti et al.

m~180...1000 MeV


m~180...1000 MeV

10 out of ~28 ensembles10 out of ~28 ensembles

NP-renormalized[Göckeler et al. 1003.5756]

NP-renormalized[Göckeler et al. 1003.5756]

[non-singlet, connected only]*

u-d u+d(*)


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u-d u+d(*)


11





u-d u+d(*)

In summary: nice data (more to come) no big surprises compares well with previous studies(QCDSF [Ohtani et al. 0710.1534], LHPC nf=2+1 [PhH et al. 0705.4295], [Bratt et al. 1001.3620]) detailed analysis (systematic uncertainties, chiral extrapolations, etc.) in the near future

In summary: nice data (more to come) no big surprises compares well with previous studies(QCDSF [Ohtani et al. 0710.1534], LHPC nf=2+1 [PhH et al. 0705.4295], [Bratt et al. 1001.3620]) detailed analysis (systematic uncertainties, chiral extrapolations, etc.) in the near future


12

Ju, Jd template figure (update)

LHPC arXiv:1001.3620

LHPC PRD `08 0705.4295

QCDSF (Ohtani et al.) 0710.1534

Goloskokov&Kroll EPJC`09 0809.4126

Wakamatsu 0908.0972

DiFeJaKr EPJC `05 hep-ph/0408173

(Myhrer&)Thomas PRL`08 0803.2775

[JLab Hall A PRL`07; HERMES JHEP`08]

[non-singlet, connected only; add. uncertainties due to chiral extrapolations, renormalization]*

**

*QCDSF/UKQCD preliminary *,1

1Sternbeck, Pleiter, Zanotti et al. 2011

lattice results fromcovariant BChPT extrapolations

[Dorati, Gail, Hemmert NPA 2008]

lattice results fromcovariant BChPT extrapolations

[Dorati, Gail, Hemmert NPA 2008]


13

Transverse momentum distributions (TMDs)in lattice QCD

[B. Musch, PhH, J. Negele, A. Schäfer, arXiv:1011.1213 (PRD); PhH, B. Musch, J. Negele, A. Schäfer, EPL 2009 (arXiv:0908.1283)B. Musch, PhD thesis arXiv:0907.2381]

motivationmotivation

hadron structure:-intrinsic k of quarks & gluons inside hadrons

-correlations between momentum, coordinate & spin DOFs

hadron structure:-intrinsic k of quarks & gluons inside hadrons

-correlations between momentum, coordinate & spin DOFs

conceptual questions related to:-TMD-(QCD)-factorization of semi-inclusive processes-probabilistic interpretation of TMDs; relation to PDFs

-relation between TMDs and GPDs-manifestly non-local operators and path geometries on the lattice

conceptual questions related to:-TMD-(QCD)-factorization of semi-inclusive processes-probabilistic interpretation of TMDs; relation to PDFs

-relation between TMDs and GPDs-manifestly non-local operators and path geometries on the lattice

phenomenology:-SIDIS, DY production

-T-odd effects & single-spin azimuthal asymmetries-factorization or correlations in x, k? (test factorization assumption)

↔ MC event generators (PYTHIA, HERWIG,.. for LHC, Tevatron)

phenomenology:-SIDIS, DY production

-T-odd effects & single-spin azimuthal asymmetries-factorization or correlations in x, k? (test factorization assumption)

↔ MC event generators (PYTHIA, HERWIG,.. for LHC, Tevatron)

HERMES (SIDIS) PRL 2009HERMES (SIDIS) PRL 2009

on Hon H

(and COMPASS PLB 2009, 2010)(and COMPASS PLB 2009, 2010)


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Definitions of TMDs and QCD factorization

solutions/proposalssolutions/proposals

Mulders et al.1996- („LO“); Anselmino et al., Radicci, Bachetta et al., … 1999- („LO“)Collins 1980- ; Collins&Metz; Collins&Hauptmann; Ji,Ma&Yuan 2004- („NLO“),...

Cherednikov&Stefanis(1980s-) 2007- („NLO“); Chay 2007 (EFT, „NLO“)Becher, Neubert 2010 (EFT) ~ „no factorization in terms of TMDs possible“

Aybat&Rogers 2011 („NnLO“?) →Collins („NnLO“?) in "Foundations of Perturbative QCD", (Cambridge University Press)

„Not yet published - available from May 2011“

Mulders et al.1996- („LO“); Anselmino et al., Radicci, Bachetta et al., … 1999- („LO“)Collins 1980- ; Collins&Metz; Collins&Hauptmann; Ji,Ma&Yuan 2004- („NLO“),...

Cherednikov&Stefanis(1980s-) 2007- („NLO“); Chay 2007 (EFT, „NLO“)Becher, Neubert 2010 (EFT) ~ „no factorization in terms of TMDs possible“

Aybat&Rogers 2011 („NnLO“?) →Collins („NnLO“?) in "Foundations of Perturbative QCD", (Cambridge University Press)

„Not yet published - available from May 2011“

SIDIS

(regularization and) removalof rapidity/light-cone

divergences

(regularization and) removalof rapidity/light-cone

divergences

gauge-invariancegauge-invariance

TMD-extractions/phenomenology

TMD-extractions/phenomenology

??

Is there is a unique, commonly acceptedexact definition of (SIDIS-)TMDs in terms of (GI)

quark- and gluon operators available?

Is there is a unique, commonly acceptedexact definition of (SIDIS-)TMDs in terms of (GI)

quark- and gluon operators available? !?


15

Transverse momentum dependent PDFs - formalism

TMDs are part of expansion in , classified according to twistTMDs are part of expansion in , classified according to twist

how can TMDs ever be accessed on an Euclidean lattice with hadrons (nearly) at rest?how can TMDs ever be accessed on an Euclidean lattice with hadrons (nearly) at rest?

define Fourier-transformedamplitudes

define Fourier-transformedamplitudes

twist-2twist-2 twist-3twist-3notation of

[Mulders, TangermannNPB 1996; Boer, Mulders

PRD 57 (1998)]

notation of[Mulders, Tangermann

NPB 1996; Boer, MuldersPRD 57 (1998)]

[Musch et al. arXiv:1011.1213 (PRD)]

example: direct -„process independent“ –

-Wilson lines

example: direct -„process independent“ –

-Wilson lines

parametrization of coordinate spacecorrelators in terms of 9 complexinvariant amplitudes

parametrization of coordinate spacecorrelators in terms of 9 complexinvariant amplitudes


16

Transverse momentum dependent PDFs – „extended“ formalism

spatially extendedU-shaped links, e.g.

spatially extendedU-shaped links, e.g.

physically more relevant

„process dependent“

or

„SIDIS“ or „DY“

physically more relevant

„process dependent“

or

„SIDIS“ or „DY“

access to T-odd structures

access to T-odd structures

Sivers, Boer-Mulders effects

Sivers, Boer-Mulders effects

parametrization of coordinate space correlators

in terms of 32 independent complex invariant amplitudes [Goeke, Metz, Schlegel

PLB 2005], e.g.

parametrization of coordinate space correlators

in terms of 32 independent complex invariant amplitudes [Goeke, Metz, Schlegel

PLB 2005], e.g.

note: the appearance of 32 structures is not a „lattice

artefact“

note: the appearance of 32 structures is not a „lattice

artefact“

the 32 invariant amplitudes can be matched on 32 independent TMDs of twist-2,3,4 [Goeke, Metz, Schlegel PLB 2005]

the 32 invariant amplitudes can be matched on 32 independent TMDs of twist-2,3,4 [Goeke, Metz, Schlegel PLB 2005]


17


18

Typical results

LHPC, Nf=2+1DW-valence

+staggerd sea


19

TMDs on the lattice

....

....renormalization (potential power divergencies)renormalization (potential power divergencies) Gaussian parametrization and

regularization of UV divergencies

Gaussian parametrization andregularization of UV divergencies


20

l2-dependence of the (renormalized) invariant amplitudes

unpolarizedunpolarized

unpolarizedunpolarized

[B. Musch, PhH, J. Negele, A. Schäfer, arXiv:1011.1213 (PRD)]


21


worm-gearworm-gearworm-gearworm-gear

long. polarizedlong. polarized

long. polarizedlong. polarized


22



23


„pretz....“„pretz....“

„pretz....“„pretz....“

transversitytransversity

transversitytransversity

compare e.g. to models:Pasquini et al PRD 2008, Bacchetta et al.,...

compare e.g. to models:Pasquini et al PRD 2008, Bacchetta et al.,...


24

Correlations in


25

Intrinsic transverse momentum densities of the nucleon

genuine effectof intrinsic transversemomentum of quarks

genuine effectof intrinsic transversemomentum of quarks

GPDs

PhH, B. Musch et al.EPL 2009, arXiv:0908.1283

PhH, B. Musch et al.EPL 2009, arXiv:0908.1283

[Boglione, Mulders PRD 60 (1999), Diehl, PhH EPJC 44

(2005)]

[Boglione, Mulders PRD 60 (1999), Diehl, PhH EPJC 44

(2005)]


26

Preliminary results for extended gauge links[calculations and results by Berni Musch (JLab)]


27

T-odd effects and transverse momentum shifts

with the Collins-Soperparameter ζ

with the Collins-Soperparameter ζ

CS evolution[CS NPB 1981,

IdJiMaYu hep-ph/0406302]

CS evolution[CS NPB 1981,

IdJiMaYu hep-ph/0406302]

move gauge link to infinity; ‚final state interactions‘ approach the light-cone; -moments

move gauge link to infinity; ‚final state interactions‘ approach the light-cone; -moments

note: limit anywaynot defined without regulator [see e.g.

BaBoDiMu 0803.0227] ↔ work at finite

note: limit anywaynot defined without regulator [see e.g.

BaBoDiMu 0803.0227] ↔ work at finite

note: regularizes rapidity divergencesnote: regularizes rapidity divergences

note: soft factors cancel out in amplitude-ratiosnote: soft factors cancel out in amplitude-ratios


28

Sivers transverse momentum shift


29

Boer-Mulders transverse momentum shift


30

Boer-Mulders transverse momentum shift – ζ-dependence

?


31

negative Boer-Mulders function for up- and down quarks in the proton and the pion[hep-lat/0612032], [0708.2249]

negative Boer-Mulders function for up- and down quarks in the proton and the pion[hep-lat/0612032], [0708.2249]

if preliminary TMD-results hold, a consistent picture could be emergingif preliminary TMD-results hold, a consistent picture could be emerging

Relation to deformed quark impact parameter distributions

chromodynamic lensing[Burkardt hep-ph/0302144, 0811.1206]

chromodynamic lensing[Burkardt hep-ph/0302144, 0811.1206]

tensor GPDs of the nucleon and pion on the lattice[QCDSF (PhH et al.) hep-lat/0612032, (BrDiHä et al.) 0708.2249]

tensor GPDs of the nucleon and pion on the lattice[QCDSF (PhH et al.) hep-lat/0612032, (BrDiHä et al.) 0708.2249]

+


32

Summary & Outlook

study different/additional link structuresstudy different/additional link structures

first direct lattice calculations of non-local operators related to TMDsfirst direct lattice calculations of non-local operators related to TMDs

ongoing promising efforts in computation of moments of GPDs on the latticeongoing promising efforts in computation of moments of GPDs on the lattice

employ larger momenta Pemploy larger momenta P

different pion masses, chiral extr.different pion masses, chiral extr.

first exploratory lattice study of „T-odd“ effectsfirst exploratory lattice study of „T-odd“ effects

investigations of systematic effects are in progress and should to be intensifiedinvestigations of systematic effects are in progress and should to be intensified

NP-renormalization, excited states, disconnected contributions,... NP-renormalization, excited states, disconnected contributions,...

with respect to invariant amplitudes/TMDswith respect to invariant amplitudes/TMDs

study η→ limit ζ-evolution

study η→ limit ζ-evolution

x (l·P)-dependence;(x,k)-correlations

x (l·P)-dependence;(x,k)-correlations

......


33

work done in collaboration with/based on results from

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, A. Sternbeck, F. Winter (Regensburg U.)

R. Horsley, J. Zanotti (Edinburgh U.)Y. Nakamura (Regensburg)

P. Rakow (Liverpool U.) D. Pleiter, G. Schierholz (DESY)

H. Stüben (ZIB); M. Ohtani (Tokyo)

(QCDSF/UKQCD)

M. Altenbuchinger, B. Musch (→JLab), W. Weise

(T39, TUM)

(additional) references: QCDSF: PoS(LAT2006)120; 0710.1534; PRL 98 222001 (2007);PRD 74:094508,2006 (hep-lat/0603028); PRL 2008 (0708.2249);

LHPC: PRL 96 502001 (2006) ; PRD 77, 094502 (2008), 0810.1933; PRD81:034507, 2010 (0907.4194); 1001.3620; PhH Phys.Rep. 2010 (0912.5483); 1011.1213

(additional) references: QCDSF: PoS(LAT2006)120; 0710.1534; PRL 98 222001 (2007);PRD 74:094508,2006 (hep-lat/0603028); PRL 2008 (0708.2249);

LHPC: PRL 96 502001 (2006) ; PRD 77, 094502 (2008), 0810.1933; PRD81:034507, 2010 (0907.4194); 1001.3620; PhH Phys.Rep. 2010 (0912.5483); 1011.1213

nucleon parton distributions from lattice qcd

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