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.
Ph. Hägler, DIS 2008 2
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
Ph. Hägler, DIS 2008 3
The nucleon spin sumrule and GPDs
Nucleon spin sumrule (Ji PRL 1997)
everything is: -gauge-invariant-scale and scheme dependent
-measurable
moments of GPDsDVCS
Ph. Hägler, DIS 2008 4
= 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
Ph. Hägler, DIS 2008 5
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
Ph. Hägler, DIS 2008 6
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
Ph. Hägler, DIS 2008 7
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
Ph. Hägler, DIS 2008 8
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
Ph. Hägler, DIS 2008 9
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
Ph. Hägler, DIS 2008 10
QCDSF/UKQCD LHPC
Isovector quark momentum fractioncomparison QCDSF/UKQCD and LHPC
substantial difference in the overall normalization of the lattice datamajor unresolved issue
Ph. Hägler, DIS 2008 11
LHPC
Gravitomagnetic form factor B20 chiral extrapolation based on covariant BChPT by Dorati, Gail, Hemmert NPA 2008
QCDSF/UKQCD
Ph. Hägler, DIS 2008 12
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
Ph. Hägler, DIS 2008 13
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
Ph. Hägler, DIS 2008 14
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
Ph. Hägler, DIS 2008 15
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
Ph. Hägler, DIS 2008 16
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
Ph. Hägler, DIS 2008 17
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
Ph. Hägler, DIS 2008 18
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)
Ph. Hägler, DIS 2008 19
Ph. Hägler, DIS 2008 20
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
Ph. Hägler, DIS 2008 21
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:
Ph. Hägler, DIS 2008 22
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
Ph. Hägler, DIS 2008 23
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
Ph. Hägler, DIS 2008 24
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
Ph. Hägler, DIS 2008 25
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
Ph. Hägler, DIS 2008 26
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
Ph. Hägler, DIS 2008 27
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
Ph. Hägler, DIS 2008 28
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
Ph. Hägler, DIS 2008 29
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
Ph. Hägler, DIS 2008 30
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
Ph. Hägler, DIS 2008 31
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
Ph. Hägler, DIS 2008 32
The GFFs A,B and C
QCDSF/UKQCD LHPC
Ph. Hägler, DIS 2008 33
QCDSF/UKQCD LHPC
Quark spin, OAM and total angular momentum
overall agreement within errors in particular Lu+d0
Ph. Hägler, DIS 2008 34
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
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