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Moments of GPDs and the Nucleon Spin Structure from Lattice QCD. Titelpage. Philipp Hägler. affiliations etc. supported by. excellence cluster universe. Overview. GPDs and generalized form factors. local operators and correlators on the lattice. lattice results on lowest moments of GPDs. - PowerPoint PPT Presentation
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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
Ph. Hägler, Exclusive 2010, JLab
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]
Ph. Hägler, Exclusive 2010, JLab
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
Ph. Hägler, Exclusive 2010, JLab
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
Ph. Hägler, Exclusive 2010, JLab
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
Ph. Hägler, Exclusive 2010, JLab
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)
Ph. Hägler, Exclusive 2010, JLab
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“
Ph. Hägler, Exclusive 2010, JLab
9
dependence
u-d u+d
u-d u+d
Ph. Hägler, Exclusive 2010, JLab
10
B20 and the anomalous gravitomagnetic moment
based on HBChPT byDiehl, Manashov, Schäfer EJPA 2006
Ando, Chen, Kao PRD 2006
based on HBChPT byDiehl, Manashov, Schäfer EJPA 2006
Ando, Chen, Kao PRD 2006
includingincluding
non-linear correlationin t and m
non-linear correlationin t and m
O(p3
)
LHPC nf=2+1 mixedarXiv:1001.3620 (updating PRD 2008)
LHPC nf=2+1 mixedarXiv:1001.3620 (updating PRD 2008)
small quark AGMsmall quark AGM
!?
[Teryaev `99-; Brodsky, Hwang et al. `00-]
Ph. Hägler, Exclusive 2010, JLab
11
C20 and the second moment of the D-term [Polyakov&Weiss `99]
LHPC nf=2+1 mixedarXiv:1001.3620 (updating PRD 2008)
LHPC nf=2+1 mixedarXiv:1001.3620 (updating PRD 2008)
includingincluding
non-linear correlationin t and m
non-linear correlationin t and m
O(p3
)
sizeable and negativesizeable and negative
!?
based on HBChPT byDiehl, Manashov, Schäfer EJPA 2006
Ando, Chen, Kao PRD 2006
based on HBChPT byDiehl, Manashov, Schäfer EJPA 2006
Ando, Chen, Kao PRD 2006
Ph. Hägler, Exclusive 2010, JLab
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)
Ph. Hägler, Exclusive 2010, JLab
13
Stability of the BChPT fit
cuts
Ph. Hägler, Exclusive 2010, JLab
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.)
Ph. Hägler, Exclusive 2010, JLab
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 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)
Ph. Hägler, Exclusive 2010, JLab
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
Ph. Hägler, Exclusive 2010, JLab
17
Contributions to the proton spin
[non-singlet, connected only; additional uncertainties due to chiral extrapolations, renormalization]
*
*
Ph. Hägler, Exclusive 2010, JLab
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]
Ph. Hägler, Exclusive 2010, JLab
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
Ph. Hägler, Exclusive 2010, JLab
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)
Ph. Hägler, Exclusive 2010, JLab
21
Quark angular momentum
emplyoing HBChPT+ results [Chen Ji PRL 2002]emplyoing HBChPT+ results [Chen Ji PRL 2002]
LHPC arXiv:1001.3620LHPC arXiv:1001.3620
BChPT-fitBChPT-fit
Ph. Hägler, Exclusive 2010, JLab
22
Lattice QCD vs relativistic quark models – QCD evolution(Wakamatsu 2005; Thomas, PRL 2008)
Ph. Hägler, Exclusive 2010, JLab
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
Ph. Hägler, Exclusive 2010, JLab
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
Ph. Hägler, Exclusive 2010, JLab
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
Ph. Hägler, Exclusive 2010, JLab
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)
LHPC arXiv:1001.3620LHPC arXiv:1001.3620
Ph. Hägler, Exclusive 2010, JLab
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
LHPC arXiv:1001.3620LHPC arXiv:1001.3620
Ph. Hägler, Exclusive 2010, JLab
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
Ph. Hägler, Exclusive 2010, JLab
29
Polarized momentum fraction
DSSV PRL 2008DSSV PRL 2008
employing HBChPT results [Chen Ji PLB 2001]employing HBChPT results [Chen Ji PLB 2001]
LHPC arXiv:1001.3620LHPC arXiv:1001.3620
Ph. Hägler, Exclusive 2010, JLab
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
LHPC arXiv:1001.3620LHPC arXiv:1001.3620
Ph. Hägler, Exclusive 2010, JLab
31
Momentum fraction of quarks in the nucleon
substantial systematic uncertaintiessubstantial systematic uncertainties
Ph. Hägler, Exclusive 2010, JLab
32
Correlations in
Correlations between momenta, positions, spins
Ph. Hägler, Exclusive 2010, JLab
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
Ph. Hägler, Exclusive 2010, JLab
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
Ph. Hägler, Exclusive 2010, JLab
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
lattice calculationsof quark spin-flip
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
Ph. Hägler, Exclusive 2010, JLab
36
BMW (Dürr et al.) Nature 2009BMW (Dürr et al.) Nature 2009
Lattice QCD propaganda
Ph. Hägler, Exclusive 2010, JLab
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
Ph. Hägler, Exclusive 2010, JLab
38
up
down
Invariant amplitudes related to quadrupole deformations(„pretzelosity“)
Ph. Hägler, Exclusive 2010, JLab
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
Ph. Hägler, Exclusive 2010, JLab
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
Ph. Hägler, Exclusive 2010, JLab
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
Ph. Hägler, Exclusive 2010, JLab
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
lattice calculationsof quark spin-flip
couplings
lattice calculationsof quark spin-flip
couplings
Ph. Hägler, Exclusive 2010, JLab
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
Ph. Hägler, Exclusive 2010, JLab
45
Overview of results for the pion charge radius
monopole fitpole+polyn.
1-loop ChPT
Ph. Hägler, Exclusive 2010, JLab
46
Nucleon anomalous magnetic moment
(anomalous) magnetic moment(anomalous) magnetic moment
Ph. Hägler, Exclusive 2010, JLab
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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