Contributions to the nucleon spin from lattice QCD calculations

Ph. Hägler, DIS 2008 1

Contributions to the nucleon spin from lattice QCD calculations

supported byPhilipp Hägler

Related talks at this conference: Xi.-D. Ji (Monday 12:25h), D. Müller (Today 14:00), M. Diehl (Wednesday 14:00h) and spin physics session on Wednesday afternoon.


Overview

Nucleon spin sum rule

brief outline of computation of nucleon matrix elements in lattice QCD

selected lattice results on• form factors of the energy momentum tensor

• spin and OAM contrubutions to the nucleon spintogether with chiral extrapolationsfrom QCDSF/UKQCD and LHPC

summary/outlook

comparison of lattice results with experiment&phenomenology

very brief intro to ChPT calculations


The nucleon spin sumrule and GPDs

Nucleon spin sumrule (Ji PRL 1997)

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

-measurable

moments of GPDsDVCS


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

gauge fields/links U

quark propagators

quarks

Lattice calculation of nucleon matrix elements

compute the path-integral numerically


APEmille at NIC/DESY Zeuthen

SGI Altix 4700 at LRZ Garching

Quellen: (http://www.lrz-muenchen.de/wir/einweihungsfeier/bildergalerie/_fotoindex.html) & Jlab webpage

Calculation of configurations and propagators

7n cluster at JLab

http://www.lrz-muenchen.de/wir/einweihungsfeier/bildergalerie/_fotoindex.html)&Jlab

http://www.lrz-muenchen.de/wir/einweihungsfeier/bildergalerie/_fotoindex.html)&Jlab


Chiral perturbation theory and chiral extrapolations

low energy effectivefield theory of QCD

baryon chiral perturbation theory (BChPT)

systematic expansion in smallmasses/momenta/energies

non-relativistic expansion

heavy baryon ChPT (HBChPT)

covariant baryon ChPT (CBChPT)

HBChPT with the Δ, small scale expansion (SSE)

expansion in 1/mN

NBernard, Hemmert, Meißner;

e.g. for gA: Hemmert, Procura, Weise PRD 2003Chen and Ji, PRL 2002

unknown couplings / low energy constants (LECs) have to be determined from experiment and/or lattice

Dorati, Gail,Hemmert NPA 2008


Lattice parameters

QCDSF/UKQCD

LHPC- domain-wall valence and staggered „Asqtad“ sea quarks („hybrid“ approach)- unquenched, Nf=2+1- only connected contributions- lattice spacing fixed using the force F1(r) - lattice spacing 0.125 fm- pion masses as low as 350 MeV- NP improved perturbative renormalization

- Wilson fermions with NP clover improvement- unquenched, Nf=2- only connected contributions- lattice spacing fixed using mN →r0=0.467fm- lattice spacings ~0.1 … 0.07 fm- pion masses as low as 350 MeV- non-perturbative renormalization

arXiv:0710.1534(proceedings)

arXiv:0705.4295(to appear in PRD)

all results transformed to MS at 4 GeV2


Isovector quark momentum fraction

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

with common

parameter

not yetfull O(p3)

HB-limit mN → ∞

HBChPT-fit

CTEQ6

LHCP PRD 2008

global simultaneouschiral 9-parameterfit to ~120 lattice

datapoints at finite t


Cross-checks

comparison of chiral fits to different regions in m

estimate (?) of O(p3)-corrections including

chiral fits agree at small and large m

bending towardschiral limit seems genuine

uncertainties dominatedby statistical errors

simple m 3 -termdoesn‘t fit lattice data well


QCDSF/UKQCD LHPC

Isovector quark momentum fractioncomparison QCDSF/UKQCD and LHPC

substantial difference in the overall normalization of the lattice datamajor unresolved issue


LHPC

Gravitomagnetic form factor B20 chiral extrapolation based on covariant BChPT by Dorati, Gail, Hemmert NPA 2008

QCDSF/UKQCD


Isosinglet B20(t) form factor(including quark anomalous gravitomagnetic moment AGM)

based on HBChPT by Diehl, Manashov, Schäfer EJPA 2006, Ando, Chen, Kao PRD 2006

including

non-linear correlationin t and m

(p3)

small quark AGM

LHCP PRD 2008


Quark angular momentum from extrapolated B20(t→0)

based on HBChPT including the Δ by Chen and Ji, PRL 2002

including e.g.

quarks carry 40% of total nucleon spin 1/2


Quark spin, OAM and total angular momentum

LHPC QCDSF/UKQCD

overall agreement within errors

see also pioneering lattice calculations by Gadiyak, Ji and Jung 2001


Towards the decomposition of the nucleon spingluon spin contribution (e.g. COMPASS/CERN)

Ji spin sumrule

lattice

„graviton-like coupling“in lattice QCD

exp/pheno

be aware of systematicuncertainties of

lattice simulations

Jlab Hall APRL 2007


HERMES 0802.2499

Towards the decomposition of the nucleon spin

substantial uncertaintiesdue to model dependence

uncertainties will bereduced in future →more HERMES data,JLab 12 GeV upgrade

be aware of systematicuncertainties of

lattice simulations


Summary&Outlook

lattice simulations provide new insights into structure of hadrons

direct access to quark angular momentum and nucleon spin sumrule

complementary to enormous experimentalefforts at e.g. HERMES, COMPASS, JLab

(near) future: studies of GPDs of the rho (M. Gürtler in collaboration with QCDSF/UKQCD)

spin structure of a spin-1 state

ChPT essential for extrapolation to the physical point


Many thanks to my collaborators

B. Bistrovic, J. Bratt, J.W. Negele, A. Pochinsky (MIT)

R.G. Edwards, D.G. Richards (JLab)K. Orginos (W&M)

M. Engelhardt (New Mexico)G. Fleming (Yale)B. Musch (TUM)

D.B. Renner (Arizona)W. Schroers (DESY Zeuthen)

(LHPC)

M. Göckeler, M. OhtaniA. Schäfer (Regensburg U.)

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

P. Rakow (Liverpool U.) D. Pleiter, Y. Nakamura,

G. Schierholz (DESY Zeuthen)W. Schroers (National Taiwan U.)

(QCDSF/UKQCD)

arXiv:0710.1534(proceedings)arXiv:0705.4295 (to appear in PRD)


Becher, Leutwyler 1999

Covariant baryon chiral perturbation theoryfor nucleon EM-FFs and FFs of the energy momentum tensor

Dorati, Gail, Hemmert NPA 2008Gail&Hemmert (tpb)

Choice of renormalization scheme

covariant BChPT includes

nucleon form factors

FFs of the energy momentum tensor

N

HBChPT

fully compatible with MS-HBChPT


Lattice parameters – LHPC/MILC

),,),(,,(p'

))((

a

m.m

pol

qres

001000

114

1~

6.1

10

350

1

momenta-sink two-

projector one-

(3.5fm) volumes in 360MeV aslow as masses pion-

GeV is spacing-lattice inverse-

16,Ls-

onscontributi connectedonly but 1,2N-

taste" of matter a" is quarks staggered of use -

smearing-HYP with formalism) hybrid"("

sea staggered Asqtad"" staggered a on fermions-wall-domain-

3

f

operator renormalization:


Lattice parameters – QCDSF/UKQCD

nscombinatio some for ionsconfigurat 1600 to up -

available and different of #largepretty -

GeV is spacing-lattice inverse-

onscontributi connectedonly -

2)(N ncalculatio unquenched-

timprovemen-clover (NP) with fermions-Wilson-

f

,

21a

ationrenormaliz operator veperturbatinon

010001000 momenta-sink three-

12

1~ 12

1~ projectors three-

467.0m using fixed spacing lattice -

available analysis volume finite for data -

2,150210

0N

),,),(,,),(,,(p'

)(),(

fmr

,unpol

# BetaVal KappaVal KappaValSea L Lt mN mPion aLatt aLattInv1 5.20 .13420 .13420 16 32 1.1053 0.584732 0.114545 1.72272 5.20 .13500 .13500 16 32 0.8646 0.414774 0.0982331 2.008763 5.20 .13550 .13550 16 32 0.6926 0.290712 0.0926403 2.130034 5.25 .13460 .13460 16 32 0.9451 0.493157 0.0985856 2.001585 5.25 .13520 .13520 16 32 0.7979 0.382066 0.0908914 2.171026 5.25 .13575 .13575 24 48 0.6061 0.255564 0.0844179 2.33757 5.25 .13600 .13600 24 48 0.5118 0.184668 0.0822183 2.400049 5.29 .13400 .13400 16 32 1.0572 0.576715 0.0970289 2.0336910 5.29 .13500 .13500 16 32 0.8350 0.420572 0.0893438 2.2086313 5.29 .13550 .13550 24 48 0.6874 0.326883 0.0839023 2.3518716 5.29 .13590 .13590 24 48 0.5656 0.23956 0.0799658 2.4676317 5.29 .13620 .13620 24 48 0.4682 0.159387 0.0774204 2.5487718 5.29 .13632 .13632 32 64 0.4202 0.137020 0.076159 2.5909819 5.40 .13500 .13500 24 48 0.7550 0.403009 0.0766579 2.5741320 5.40 .13560 .13560 24 48 0.6240 0.312317 0.073186 2.6962421 5.40 .13610 .13610 24 48 0.5098 0.220812 0.069556 2.8369522 5.40 .13640 .13640 24 48 0.414 0.149935 0.0675734 2.92019


Generalized form factors and moments of GPDs

etc.

)(4)(

)(4)(

),(),(

202

20

1

1

202

20

1

1

102

1

1

101

1

1

tCtB,t)E(x, xdx

tCtA,t)H(x, xdx

tB(t)F,t) E(x,dxtA(t)F,t) H(x,dx

moments of GPDs

elementsmatrix of

moments Mellin 11

1

n

-

xdx

2/)'(

)'( 22

PPP

PPt

12118743322

4422122

54322

,,,,,nn/i

,,,n/

,,,n

: tensor

: vector axial

: vector

5

)()(

)(2

)()'()0()0('

20

020

1

1

11

PUtCm

tBm

PitAPPUPqiDqP n

# of GFFs

for n=2


Computation of moments of GPDs on the lattice

disconnected contributions 0 ,1det dynamical) (full, fnUDunquenched

x

,'P P

very expensive,not included so far

drop out for u-d

finite lattice spacing finite volume large quark masses

systematic uncertainty in isosinglet quantities


020

2020

du

dudu

C

AA

0

0

20

2020

du

dudu

C

BB

compatible withlarge Nc limit – see e.g. Goeke, Polyakov

and Vanderhaeghen PiPaNP 2001

in summary

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

Overview 1: The GFFs A,B and C

)(4)(

)(4)(

202

20

1

1

202

20

1

1

tCtB,t)E(x, xdx

tCtA,t)H(x, xdx


Quark spin and OAM contributions to the nucleon spin

0)0(2

1

2

120 dududududu

qdu

q BxJL dududu xB and 0)0(20

MeV350 at m

HERMES 2007results for Δ


Quark spin and OAM contributions to the nucleon spin

2

1 of %30 u

qdq LL

01.002.0

01.026.0

dq

uq

J

J

MeV350 at m

HERMES 2007results for Δ


At finite momentum transfer

0.1 0.2 0.3 0.4 0.5 0.6m

2GeV20.05

0.1

0.15

0.2

0.25

A02udt42.0

VeG

2

0.1 0.2 0.3 0.4 0.5 0.6m

2GeV20.05

0.1

0.15

0.2

0.25

A02udt42.0

VeG

2


Example 2: Isosinglet quark momentum fractionchiral extrapolation based on CBChPT by Dorati, Gail, Hemmert 2007

0.1 0.2 0.3 0.4 0.5 0.6m

2GeV20.1

0.2

0.3

0.4

0.5

0.6

A02u

dt.0

VeG

2

0.1 0.2 0.3 0.4 0.5 0.6m

2GeV20.1

0.2

0.3

0.4

0.5

0.6

A02u

dt.0

VeG

2

warning: only a part of CBChPT(p3)-corrections are known

HB-limit mN → ∞HBChPT-fit

with common

parameter

global simultaneouschiral 9-parameterfit to ~120 lattice

datapoints at finite t

CTEQ6


At finite momentum transfer

0.1 0.2 0.3 0.4 0.5 0.6m

2GeV20.1

0.2

0.3

0.4

0.5

0.6

A02udt42.0

VeG

2

0.1 0.2 0.3 0.4 0.5 0.6m

2GeV20.1

0.2

0.3

0.4

0.5

0.6

A02udt42.0

VeG

2


Example 2: Isosinglet C20(t) form factorchiral extrapolation based on CBChPT by Dorati, Gail, Hemmert 2007

non-lineardependence

on (t,m)

(p3)(p3)

warning: only a part of CBChPT (p3)-corrections are known

including


The GFFs A,B and C

QCDSF/UKQCD LHPC


QCDSF/UKQCD LHPC

Quark spin, OAM and total angular momentum

overall agreement within errors in particular Lu+d0


Conclusions

two different dynamical lattice simulations one message?!

02.000.0

2%44ˆ02.022.0

2%44ˆ03.022.0

d

u

du

J

J

J

1 of

1 of

21%ˆ04.020.0 of 40 ud LL

04.00.0 duL

02.052.0)0(

01.016.0)0(

20

20

dudu

dudu

xA

xA

04.042.0

26.1

du

Adu g

(CTEQ6)

(beta-decay & HERMES PRL 2007)

disclaimer: additional uncertainties arising from-missing disconnected contributions,

- discretization effects,- finite size effects,

- chiral extrapolation (in particular for B20)

1.03.0)0(

02.00.0)0(

20

20

du

du

C

C

1.01.0)0(

06.027.0)0(

20

20

du

du

B

B

Contributions to the nucleon spin from lattice QCD calculations

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