Titelpage

Moments of GPDsand the Nucleon Spin Structure

from Lattice QCD

Philipp Hägler

supported by

excellence cluster universe

Ph. Hägler, Exclusive 2010, JLab

2

Overview

GPDs and generalized form factorsGPDs and generalized form factors

local operators and correlators on the latticelocal operators and correlators on the lattice

lattice results on lowest moments of GPDslattice results on lowest moments of GPDs

chiral extrapolationschiral extrapolations

decomposition of the nucleon spindecomposition of the nucleon spin

Summary and challengesSummary and challenges


3

GPDs and hadron structure

center of momentum

Ji‘s nucleon spin sum rule (X. Ji, PRL 1997)Ji‘s nucleon spin sum rule (X. Ji, PRL 1997)

everything is: -gauge-invariant

-scale & scheme dependent-measurable

everything is: -gauge-invariant

-scale & scheme dependent-measurable

Decomposition of the nucleon spin

Tomography [Burkardt PRD`00;Ralston, Pire PRD‘02]

[Diehl,PhH EPJC`05; QCDSF PRL`07`08]


4

DVCS (– factorization)[Müller at el. `94-, Radyushkin, Ji `97-, Collins

`99,`01]

bilocal on light-conefirst exploratory studies of non-local

couplings on the lattice related to TMDs

[Musch(Fr 2pm), PhH, Schäfer, Negele]

first exploratory studies of non-localcouplings on the lattice

related to TMDs[Musch(Fr 2pm), PhH, Schäfer, Negele]

for the time being:local couplings

for the time being:local couplings

GPDs and non-local operators

Higher moments and local operators

... of matrix elements... of matrix elements

…of GPDs…of GPDs

Ji&Lebed PRD 2000Ph.H. PLB 2004

Ji&Lebed PRD 2000Ph.H. PLB 2004

GFFsGFFs


5

Local quark operators on the lattice

leading twist-2 leading twist-2

operator renormalizationoperator renormalization

lattice perturbation theorylattice perturbation theory non-perturbativelynon-perturbatively

operator mixing under renormalizationoperator mixing under renormalizationin singlet sectorin singlet sector

non-singlet sectornon-singlet sector H(4)H(4)

gluon-opsgluon-ops

discretized operatorson a Euclidean space-time lattice

discretized operatorson a Euclidean space-time lattice LorentzO(4) H(4)LorentzO(4) H(4)

n=2 n=2


6

gauge fields/links U

quark propagators

quarks

Correlation functions on the lattice

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

products of quark propagatorsproducts of quark propagators


7

Lattice simulation details

operatorrenormalization:

# of „measurements“ increased byfactor 8 compared to PRD 77 094502 (2008)

# of „measurements“ increased byfactor 8 compared to PRD 77 094502 (2008)

- mixed action approach: DW fermions on a Asqtad staggered sea for Nf=2+1; including HYP-smearing- Ls=16, mres0.1mq

- lattice spacing a ~ 0.124 fm- volumes of ~2.5 and ~3.5 fm3

- two sink momenta P‘=(0,0,0), (-1,0,0)

- mixed action approach: DW fermions on a Asqtad staggered sea for Nf=2+1; including HYP-smearing- Ls=16, mres0.1mq

- lattice spacing a ~ 0.124 fm- volumes of ~2.5 and ~3.5 fm3

- two sink momenta P‘=(0,0,0), (-1,0,0)

ongoing efforts within LHPC based on DW fermions (RBC/UKQCD) and improved Wilson fermions (BMW)ongoing efforts within LHPC based on DW fermions (RBC/UKQCD) and improved Wilson fermions (BMW)


8

n=2 - A, B, C - Form factors of the energy momentum tensor

seems to be compatible with large Nc limit – see e.g. Goeke, Polyakov, Vanderhaeghen PiPaNP 2001

disconnected contributionsare not included↔only u-d is „exact“


9

dependence

u-d u+d

u-d u+d


10

B20 and the anomalous gravitomagnetic moment

based on HBChPT byDiehl, Manashov, Schäfer EJPA 2006

Ando, Chen, Kao PRD 2006



includingincluding

non-linear correlationin t and m


O(p3

)

LHPC nf=2+1 mixedarXiv:1001.3620 (updating PRD 2008)


small quark AGMsmall quark AGM

!?

[Teryaev `99-; Brodsky, Hwang et al. `00-]


11

C20 and the second moment of the D-term [Polyakov&Weiss `99]



includingincluding



O(p3

)

sizeable and negativesizeable and negative

!?






12

Global, simultaneous chiral extrapolation of A, B, C

with common

parameter

with common

parameter

chiral extrapolation based on covariant BChPT by Dorati, Gail, Hemmert NPA 2008chiral extrapolation based on covariant BChPT by Dorati, Gail, Hemmert NPA 2008

only quark line connected contributionsonly quark line connected contributions

LHPC nf=2+1 mixed; arXiv:1001.3620(updating PRD 2008, 0810.1933)

LHPC nf=2+1 mixed; arXiv:1001.3620(updating PRD 2008, 0810.1933)


13

Stability of the BChPT fit

cuts


14

Quark angular momentum

from covariant BChPT extrapolationsfrom covariant BChPT extrapolations

LHPC arXiv:1001.3620LHPC arXiv:1001.3620

preliminary LHPC DW (Syritsyn et al.)preliminary LHPC DW (Syritsyn et al.)


15

preliminary LHPC DW (Syritsyn et al.) preliminary LHPC DW (Syritsyn et al.)

Quark spin and OAM

relativistic quark modelsrelativistic quark models


LHPC nf=2+1 mixedarXiv:1001.3620 (updating PRD 2008)LHPC PRD 2003LHPC PRD 2003

LHPC PRD D 77, 094502 (2008)LHPC PRD D 77, 094502 (2008)


16

Nucleon spin structure and spin sum rule

pioneering lattice calculations by Gadiyak, Ji and Jung in 2001pioneering lattice calculations by Gadiyak, Ji and Jung in 2001

LHPC nf=2+1 mixedarXiv:1001.3620

LHPC nf=2+1 mixedarXiv:1001.3620


17

Contributions to the proton spin

[non-singlet, connected only; additional uncertainties due to chiral extrapolations, renormalization]

*

*


18

Ju, Jd template figure

LHPC arXiv:1001.3620 (this work)

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]


19

Conclusions and outlook

decomposition of the nucleon spindecomposition of the nucleon spin„gravitational“ coupling„gravitational“ coupling

lattice QCD a great tool to study all sorts of local couplingslattice QCD a great tool to study all sorts of local couplings

transversity,transverse spin structure of hadrons

transversity,transverse spin structure of hadronsspin-flip couplingspin-flip coupling

(axial-) vector coupling(axial-) vector coupling form factors,magnetic moments,…

form factors,magnetic moments,…

form factors (radii, magnetic moments, etc.)[H.-W. Lin Wed ~3pm]: systematic uncertaintiesform factors (radii, magnetic moments, etc.)[H.-W. Lin Wed ~3pm]: systematic uncertainties

Challenges

disconnected diagrams; strange quark contributions; gluon operators; operator mixing;… disconnected diagrams; strange quark contributions; gluon operators; operator mixing;…

complementary to experimental efforts at JLab, HERMES, COMPASS; phenomenological and model studiescomplementary to experimental efforts at JLab, HERMES, COMPASS; phenomenological and model studies


20

as always, I am indebted to my collaborators

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, Th. Hemmert, A. Schäfer (Regensburg U.)

M. Gürtler (TU München)R. Horsley, J. Zanotti (Edinburgh U.)

Y. Nakamura (DESY Zeuthen) P. Rakow (Liverpool U.)

D. Pleiter, G. Schierholz (DESY)H. Stüben (ZIB)

(QCDSF/UKQCD)

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

(T39, TUM)

References: QCDSF PoS(LAT2006)120, 0710.1534, PRL 98 222001 (2007), PRL 2008 (0708.2249), Brömmel et al EPJC 2007; LHPC PRD 77, 094502 (2008), 0810.1933; 1001.3620;

Diehl&Hägler EPJC hep-ph/0504175; Musch et al. 0811.1536; Musch arXiv:0907.2381; PhH, Musch et al. EPL 2009 (arXiv:0908.1283)

PhH Phys.Rep. 2010 (0912.5483)


21

Quark angular momentum

emplyoing HBChPT+ results [Chen Ji PRL 2002]emplyoing HBChPT+ results [Chen Ji PRL 2002]


BChPT-fitBChPT-fit


22

Lattice QCD vs relativistic quark models – QCD evolution(Wakamatsu 2005; Thomas, PRL 2008)


23

Lattice QCD vs relativistic quark models – QCD evolution

0 . 2 0 . 3 0 . 4 0 . 5

0 . 4

0 . 2

0 . 0

0 . 2

0 . 4

0 . 6

J g

2L u d

relativistic quark modelrelativistic quark model

(Wakamatsu 2005; Thomas, PRL 2008)

in coll. with LHPC andM. Altenbuchinger, W. Weise (TUM)

in coll. with LHPC andM. Altenbuchinger, W. Weise (TUM)

lattice + evolutionlattice + evolution


24

Lattice in contradiction with well-known (analytical and model) results?

requires non-vanishing quark OAM L0 requires non-vanishing quark OAM L0

non-zero Sivers effectnon-zero Sivers effect

Brodsky, Hwang et al NPB 2001, PLB 2002Burkardt, Hwang PRD 2004

Brodsky, Hwang et al NPB 2001, PLB 2002Burkardt, Hwang PRD 2004

relativistic quark modelsrelativistic quark models


25

outdated

old preliminaryunpublished

proceedingsarXiv:0710.1534

Published PRD 2008

arXiv:0705.4295 disclaimer: JLab band

from integral overVGG model, constrained

at a single x=-point

Comparison with phenomenology, previous lattice studiesfrom Hall A PRL 2007;

0709.0450

from Hall A PRL 2007; 0709.0450

this workarXiv:1001.3620


26

Isovector axial vector coupling constant(required for L=J-Σ/2)

emplyoing SSE (HBChPT+) results [Procura, Hemmert, Musch, Weise PRD 2007, QCDSF PRD 2006]emplyoing SSE (HBChPT+) results [Procura, Hemmert, Musch, Weise PRD 2007, QCDSF PRD 2006]

compare to LHPC PRL 96 502001 (2006) compare to LHPC PRL 96 502001 (2006)



27

Isosinglet quark spin fraction(required for L=J-Σ/2)

employing HBChPT by Diehl, Manashov, Schäfer EJPA 2006; Ando, Chen, Kao PRD 2006

HERMES PRD 2007HERMES PRD 2007



28

Covariant chiral perturbation theory for A, B, C

with common

parameter

with common

parameter

Covariant BChPT calculation by Dorati, Gail, Hemmert NPA 2008Covariant BChPT calculation by Dorati, Gail, Hemmert NPA 2008

Including the dependence on the squared momentum transfer t Including the dependence on the squared momentum transfer t similar results for B, Csimilar results for B, C


29

Polarized momentum fraction

DSSV PRL 2008DSSV PRL 2008

employing HBChPT results [Chen Ji PLB 2001]employing HBChPT results [Chen Ji PLB 2001]



30

HBChPT-fit

compared to LHPC PRD 77 094502 (2008) compared to LHPC PRD 77 094502 (2008)

Form factors of the energy momentum tensorisovector quark momentum fraction



31

Momentum fraction of quarks in the nucleon

substantial systematic uncertaintiessubstantial systematic uncertainties


32

Correlations in

Correlations between momenta, positions, spins


33

Reminder: Generalized mean square radii of the nucleoncorrelations in x and b

strong correlations in x and b strong correlations in x and b no factorization of GPDs in x and t no factorization of GPDs in x and t

LHPC nf=2+1 mixed preliminary(updating PRD 2008)

LHPC nf=2+1 mixed preliminary(updating PRD 2008)xb

yb

bu

d

zP

z

zxP

u


34

no visible correlations in l P and l2no visible correlations in l P and l2

≈ factorization of tmdPDFs in x and k ≈ factorization of tmdPDFs in x and k

Transverse momentum dependent PDFscorrelations in x and k

Musch et al. nf=2+1 mixedtbp and PoS LC2008

Musch et al. nf=2+1 mixedtbp and PoS LC2008


35

Spin structure of the pion

Is the pion spinstructure trivial?

Is the pion spinstructure trivial?pion spin sumrulepion spin sumrule

the pion has a non-trivial transverse spin structure!the pion has a non-trivial transverse spin structure!

xb

yb

bu

d

zP

z

zxP

up-quarks in a +

quark transversespin in x-direction

lattice calculationsof quark spin-flip

couplings


couplingsQCDSF nf=2 Clover, PRL 2008QCDSF nf=2 Clover, PRL 2008

but is non-zero?

QCDSF nf=2 Clover PRL 2008QCDSF nf=2 Clover PRL 2008


36

BMW (Dürr et al.) Nature 2009BMW (Dürr et al.) Nature 2009

Lattice QCD propaganda


37

Lattice QCD propaganda

Davies, Lepage et al. PRL 2008Davies, Lepage et al. PRL 2008

Davies, Lepage et al. PRL 2004Davies, Lepage et al. PRL 2004


38

up

down

Invariant amplitudes related to quadrupole deformations(„pretzelosity“)


39

Intrinsic transverse momentum densities of the nucleon

Musch et al. tbpMusch et al. tbp

up down

up down

LC quark model:Pasquini et al PRD 2008

LC quark model:Pasquini et al PRD 2008


41

rotational symmetryrotational symmetry

translation invariancetranslation invariance

conservation of angular momentumconservation of angular momentum

conservation of momentumconservation of momentum

Form factors of the energy momentum tensor and fundamental sumrules

graviton-coupling spin-2 couplinggraviton-coupling spin-2 coupling

momentum sumrulemomentum sumrule

Ji‘s nucleon spin sum ruleJi‘s nucleon spin sum rule

everything is: -gauge-invariant-scale and scheme dependent

-measurable

everything is: -gauge-invariant-scale and scheme dependent

-measurable

vanishing of the anomalousgravitomagnetic moment

vanishing of the anomalousgravitomagnetic moment


42

Transversely polarized quarks in transversely polarized nucleons

probability densityfor transversely polarized quarks in

a transversely polarized proton

probability densityfor transversely polarized quarks in

a transversely polarized proton

Diehl / PhH EPJC 2005Diehl / PhH EPJC 2005

multipole-expansionmultipole-expansion

monopolemonopole dipoledipole quadrupolequadrupole


43

Transverse spin densities in the proton

xb

yb

bu

d

zP

z

zxP

u

up

down

charge distribution

1.2 fm

up

down

quark transversespin in x-direction

strongly deformed transverse spin densities

„Femto-photography“of quarks in the proton(Pire&Ralston PRD 2002)

„Femto-photography“of quarks in the proton(Pire&Ralston PRD 2002)

QCDSF nf=2 CloverPRL 2007

QCDSF nf=2 CloverPRL 2007


couplings


couplings


44

Charge density in the +-baryon (spin-3/2)

Alexandrou et al.nf=0(2) Wilson, PRD 2009

Alexandrou et al.nf=0(2) Wilson, PRD 2009


45

Overview of results for the pion charge radius

monopole fitpole+polyn.

1-loop ChPT


46

Nucleon anomalous magnetic moment

(anomalous) magnetic moment(anomalous) magnetic moment


47

Local quark operators on the lattice

leading twist-2 leading twist-2

discretized operatorson a Euclidean space-time lattice

discretized operatorson a Euclidean space-time lattice LorentzO(4) H(4)LorentzO(4) H(4)

operator renormalizationoperator renormalization

lattice perturbation theorylattice perturbation theory non-perturbativelynon-perturbatively

operator mixing under renormalizationoperator mixing under renormalizationin singlet sectorin singlet sector

non-singlet sectornon-singlet sector

n=2 n=2

H(4)H(4)

gluon-opsgluon-ops

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