Understanding nucleon structure using lattice simulations

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by  Wolfram Schroers

EPJ manuscript No.
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Understanding nucleon structure using lattice simulations
Recent progress on three different structural observables
Wolfram Schroers
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John von Neumann-Institutfu¨r Computing NIC/DESY,15738 Zeuthen,Germany
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2 Received: date/ Revised version: date

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a Abstract. This review focuses on the discussion of three key results of nucleon structure calculations on
J thelattice.Thesethreeresultsarethequarkcontributiontothenucleonspin,Jq,thenucleon-∆transition
4 form factors, and the nucleon axial coupling, gA. The importance for phenomenology and experiment is
discussed and therequirementsfor futuresimulations are pointed out.

1 Preprint number:DESY06-194
v
3
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0 PACS. 12.38.Gc Lattice gauge theory,nucleon structure
1
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7 1 Introduction effective theoryatfinite lattice spacing,andthe existence
0
of the continuum limit is crucial. Furthermore, usually
/
t staggered-typequarks are being used for the sea with the
a In recent years lattice gauge theory has become a ma-
l tureandreliablewaytoinvestigatethestructureofstrong square-rootbeingtakenofthedeterminant.Itisnotclear
-
p interactions. It provides a model-independent way to do if the procedure of taking the square rootcommutes with
e calculationsinQCD.However,contemporarylatticecom- takingthecontinuumlimit,seee.g.[5]forarecentreview.
h Finally, the matching of sea- and valence-quark masses is
putationsbecomeextremelycostlyatquarkmassescorre-
: prescription-dependent, and particular choices may give
v spondingtopionmassesbelow500MeV.Nature,however,
i has chosen the pion mass to be only 140 MeV. The light- rise to additional possibly large O(a2) artifacts [6].
X
In this reviewwe focus onthree observableswith rele-
ness of the pseudoscalar mesons is due to the mechanism
r vancetophenomenologicalandexperimentalapplications.
a of spontaneous chiral symmetry breaking.If, however,we
Thefirstoneisthequarkcontributiontothenucleonspin,
can investigate only the regime of heavy quarks, where
J . The second one is the transition form factors of the
chiral symmetry is broken explicitly by the quark mass, q
nucleon-∆ transition. The third one is the nucleon axial
we might not describe physics accurately at light quark
coupling, g . The former two of these quantities have so
masses. A
far been understood qualitatively, but a precise matching
Toaddressandovercomethischallenge,threedifferent
between the light quark regime and the lattice — possi-
procedures have been proposed and are actively pursued:
bly by chiral perturbation theory or an effective model of
(i) Pushing existing simulations with Wilson-type quarks
the strong interaction like [7] — still remains to be done.
down to smaller quark masses by relying on improved al-
For the latter observable it has been shown that lattice
gorithms and faster computers [1], (ii) using a hybrid ac-
data can in fact be consistent with experiment when fit-
tionapproachby using differentformulationsfor sea-and
ting it using the leading logarithmic chiral perturbation
valence-quarks [2], and (iii) doing simulations using dy-
theoryexpression.This achievementmarksamilestonein
namical Ginsparg-Wilson formulations, such as Domain-
the field of nucleon structure.
Wall fermions [3] or Overlap fermions [4]. The last ap-
proach is certainly the most challenging and demanding
one since the entire parameter space has to be explored
again.Thisappliesalsotoheavyquarks,aregimeinwhich 2 Quark contribution to nucleon spin
Ginsparg-Wilsonfermionsareabout30to100timesmore
expensive than standard Wilson-type fermions. The quark contribution to the spin of the nucleon has
The hybrid action ansatz is an excellent compromise been under intense scrutiny after the observation that
between quark mass and performance, but suffers from only about(20±15)%of the nucleonspin arisesfromthe
conceptualproblems.Firstofall,thehybridtheorybreaks quark spin [8]. Recently, it has been realized how the use
unitarityatfinitelatticespacing.Thus,itcannotactasan ofGPDs[9,11,10]providesthemeanstodirectlycompute
2 Wolfram Schroers: Understandingnucleon structure using lattice simulations
the quark contributionto the nucleon spin via the energy takentoresolvethisimportantquestion.Wecanconclude,
momentum tensor [10] however, that the technology and understanding of how
to compute these matrix elements are available and can
J = lim(Au+d(t)+Bu+d(t)) . (1)
q t→0 20 20 be deployed easily once sufficiently light pion masses are
available.
The virtuality t is given by t ≡ (p′ −p)2, where p′ and
p are the nucleon’sincoming andoutgoingmomenta.The
generalizedformfactors,Au2+0d(t) and B2u0+d(t), show up in 3 N → ∆ transition form factors
the parameterization of the nucleon’s energy-momentum
tensor. For further details and the exact definition con-
A key question is whether the baryon states of QCD are
sult [10]. The challenge is to understand which fraction
spherical or deformed. Although the nucleon is easily ac-
of the nucleon spin, J = 1/2, arises from the quark
N cessible in exclusive and inclusive scattering experiments,
spin, 1/2Σ , the quark orbital angular momentum, L ,
q q it cannot have a spectroscopic quadrupole moment since
and which fraction comes from gluon contributions, J :
g ithasspinJ =1/2.Theexcitedstateswithspin3/2and
N
above can have a quadrupole moment, but these are not
J =1/2=J +J =1/2Σ +L +J . (2)
N q g q q g
easily accessible in experiments. The only way to learn
The value of Σ has been known before [12]. The new in- about deformations of the low-lying baryon spectrum is
q
gredient is the ability to directly calculate J , and thus to consider transitions between the nucleon and the first
q
also Lq. To this end, there is no experimental determina- excited state, the ∆(1232) resonance. Experimentally, a
tion of that quantity. The first computation of Jq on the flurry of activity has recently lead to several important
lattice has been done in [12]. This calculation only uti- and exciting results [17].
lizesquenchedWilsonfermions,butfeaturesacalculation The nucleon-∆ transition can be parameterized us-
of the disconnected contribution using noisy estimators. ing three form factors — the dominant magnetic dipole
Alatercalculation[13]calculatesallgeneralizedformfac- form factor, GM1, the electric quadrupole, GE2, and the
torsintheenergy-momentumtensorseparatelyandatthe Coulomb quadrupole, GC2. Should the nucleon-∆ system
same time a publication [14] features full QCD and intro- be deformed, the latter two form factors will not vanish.
ducesanimprovedtechnologytoextractformfactorsfrom Should the system be spherical, only the magnetic dipole
matrixelements.HighermomentsofGPDshavealsobeen form factor will be non-zero.
computed [15]. On the lattice, publications reporting the successful
As of today, the understanding gained from the world computationofthesetransitionformfactorsare[18].This
of pions weighing 500 MeV and beyond is that the quark setofcalculationsusedunquenchedWilsonfermionswith
contributiontothenucleonspinisabout70%,allofwhich pionmassesbeyond600MeVandquenchedWilsonferm-
comes from the quark spin alone. The remaining 30% ions with pion masses larger than 370 MeV. Later it has
comes from the gluons. The quark orbital angular mo- been attempted to apply these techniques also for hybrid
mentum is negligibly small due to a cancellation between actions [19], but to this end the statistical error bars on
the contributions of u- and d-quarks [2]. the quadrupole form factors turn out to be too large.
Thisresultdiffersfromthefindingoutlinedabovewhich Fromthesestudiesithasbeenclearlyestablished,that
indicates that this quantity can be expected to substan- the nucleon-∆ system is indeed deformed. The sign and
tially depend on the pion mass. The cancellation of the theorderofmagnitudeofthequadrupoleformfactorsGE2
orbital angular momentum for u- and d-quarks is an in- andGC2 wasextractedsuccessfully.Theheavypionworld
terestingqualitative feature.The insightthatthe nucleon in fact is similar to Nature for these observables.
in the heavy pion world receives a larger fraction of its However,theextrapolationtothephysicalpionmasses
spin from quarks rather than gluons is compatible with yields an inconsistency for GC2 at values below Q2 <
expectations from the non-relativistic quark model, but 0.2GeV2. This discrepancy has been addressed recently
the exact interpolation between the heavy quark and the inthe frameworkofchiralperturbationtheory[20].Itap-
light quark regime can give further insight into how the pearsplausiblethatthediscrepancyinfactarisesfromthe
strong interaction operates. inadequacy of a linear chiral extrapolation — it still re-
The extrapolation to the chiral regime, however, has mainstobeseeniflatticedataatsmallerpionmassescan
not yet been possible and hence a precise quantitative indeedverify the pion mass dependence suggestedin[20].
matching with Nature has not yet been established. Al-
though Ref. [16] suggests a rather flat expression it is yet
unclear whether the same straight line is to be used for 4 Nucleon axial coupling gA
the light quark regime as the one fitting the simulations.
In this situation it is inevitable to perform similar calcu- The investigationofthe nucleonaxialcoupling has along
lations at smaller pion masses before a matching between history on the lattice, see [21] for recent reviews. Several
lattice and small-scale expansion schemes can be estab- groups have performed investigations using a wide array
lished and a definitive prediction from the lattice can be of different lattice actions, spacings, volumes, and pion
provided. masses.
Further investigations from several groups are under- Recently, two independent papers [22] and [23] have
way and all three different paths outlined in Sec. 1 are appeared showing how current lattice data can in fact
Wolfram Schroers: Understandingnucleon structure using lattice simulations 3
so far, we believe that the upcoming generation of lattice
calculations will be able to settle the debate.
This work was supported in part by the DFG, con-
1.4 tract FOR 465 (FG Gitter-Hadronen-Ph¨anomenologie),
andinpartbythe EUIntegratedInfrastructureInitiative
1.2 Hadron Physics (I3HP), contract number RII3-CT-2004-
506078.
1
g 0.8
A
0.6 LHPC/MILC References
LHPC/SESAM
RBCK
0.4 QCDSF/UKQCD 1. K. Jansen et al., these proceedings; M. G¨ockeler et al.,
QCDSF/UKQCD (small V)
0.2 Experiment arXiv:hep-lat/0610066.
2. J. W. Negele et al., Nucl. Phys. Proc. Suppl. 128
00 0.2 0.4 0.6 0.8 (2004) 170; D. B. Renner et al. [LHP Collaboration],
m 2 (GeV2) Nucl. Phys. Proc. Suppl. 140 (2005) 255; Ph. H¨agler,
π
J. W. Negele, D. B. Renner, W. Schroers, T. Lippert
and K. Schilling [LHPC Collaboration], Eur. Phys. J. A
Fig. 1. FullQCDcomputationsofthenucleonaxialcoupling,
24S1 (2005) 29; R. G. Edwards et al. [LHPC Collabora-
gA. The line shows the fit of the leading logarithmic χPT ex- tion], PoS LAT2005 (2006) 056; R. G. Edwards et al.,
pression to the hybrid lattice data from the LHPC collabora-
arXiv:hep-lat/0610007.
tion. Results from other groups are plotted, but not included
3. R.Tweedieetal.[UKQCDandRBCCollaborations],PoS
in the fit. Figure taken from Ref. [22], QCDSF data updated
LAT2005 (2006) 096.
from Ref. [23].
4. N. Cundy, S. Krieg, A. Frommer, T. Lippert and
K. Schilling, Nucl. Phys. Proc. Suppl. 140, 841 (2005);
N. Cundy, S. Krieg and T. Lippert, PoS LAT2005, 107
be combined with chiral perturbation theory to arrive at
(2006); S. Sch¨afer, arXiv:hep-lat/0609063.
the experimental value. Figure 4 shows the application of
5. S. Du¨rr, PoS LAT2005, 021 (2006).
the leading logarithmic expression from χPT to the hy-
6. O.B¨ar,C.Bernard,G.RupakandN.Shoresh,Phys.Rev.
brid data computed by the LHPC collaboration in [22].
D 72, 054502 (2005).
The gray-shaded error band shows the error arising from
7. K. Goeke, J. Ossmann, P. Schweitzer and A. Silva, Eur.
statistical uncertainties only. The fit yields quantitative Phys. J. A 27, 77 (2006).
agreement with the experimental value. 8. R. D. Ball, S. Forte and G. Ridolfi, Phys. Lett. B 378
However, the applicability of the leading order chiral (1996) 255.
perturbation theory expression to the pion masses avail- 9. D. Mu¨ller, D. Robaschik, B. Geyer, F. M. Dittes and
able has been questioned in [24]. The flat behavior at J. Horejsi, Fortsch. Phys.42 (1994) 101.
pion masses beyond 300 MeV is attributed to fine-tuning 10. X. D.Ji, Phys. Rev.Lett. 78 (1997) 610.
between different terms in the expansion. On the other 11. A. V.Radyushkin,Phys. Rev.D 56 (1997) 5524.
hand, the expansion can still be consistent with experi- 12. N. Mathur, S. J. Dong, K. F. Liu, L. Mankiewicz and
ment when applied to lattice calculations employing pion N. C. Mukhopadhyay,Phys.Rev.D 62 (2000) 114504.
masses as large as 600 MeV [23]. It is perhaps fair to say 13. M. G¨ockeler, R. Horsley, D. Pleiter, P. E. L. Rakow,
that the exact range of applicability of χPT is under de- A. Sch¨afer, G. Schierholz and W. Schroers [QCDSF Col-
laboration], Phys. Rev.Lett. 92 (2004) 042002.
bate. Nonetheless, the striking agreement between the fit
14. P.H¨agler,J.Negele,D.B.Renner,W.Schroers,T.Lippert
oflatticedataandtheexperimentalvaluemarkanimpor-
and K. Schilling [LHPC collaboration], Phys. Rev. D 68,
tant milestone for the lattice treatment of nucleon struc-
034505 (2003).
ture.
15. P.H¨agler,J.W.Negele,D.B.Renner,W.Schroers,T.Lip-
pert and K. Schilling [LHPC Collaboration], Phys. Rev.
Lett. 93 (2004) 112001.
5 Summary 16. J. W. Chen and X. d. Ji, Phys. Rev. Lett. 88 (2002)
052003.
17. G.Blanpiedet al.[LEGSCollaboration], Phys.Rev.Lett.
We have given three examples of recent lattice calcula-
76, 1023 (1996); R. Beck et al., Phys. Rev. C 61, 035204
tions which are of great interest to both phenomenolo- (2000); C. Mertz et al., Phys. Rev. Lett. 86, 2963 (2001);
gists and experimentalists alike. The limiting factor of all N. F. Sparveris et al. [OOPS Collaboration], Phys. Rev.
these lattice results, however,is their limitation to rather Lett. 94, 022003 (2005); K. Joo et al. [CLAS Collabora-
large quark masses. Currently, the question of chiral ex- tion],Phys.Rev.Lett.88,122001(2002);A.M.Bernstein,
trapolationsisunderdebateandtheapplicabilitydepends Eur. Phys. J. A 17, 349 (2003).
stronglyontheobservable.Whilesomegroupssuccessfully 18. C. Alexandrou et al., Phys. Rev. D 69, 114506 (2004);
applyfitstopionmassesaslargeas600MeV,othergroups C. Alexandrou, Ph. de Forcrand, H. Neff, J. W. Negele,
believe that pion masses lower than 300 MeV are essen- W.SchroersandA.Tsapalis, Phys.Rev.Lett.94,021601
tial. While the latter mass regime has not been reachable (2005).
4 Wolfram Schroers: Understandingnucleon structure using lattice simulations
19. C. Alexandrou et al.,PoS LAT2005, 091 (2006).
20. V.PascalutsaandM.Vanderhaeghen,Phys.Rev.Lett.95
(2005) 232001.
21. W.Schroers, Nucl. Phys.A 755 (2005) 333; W.Schroers,
Nucl.Phys. Proc. Suppl.153 (2006) 277.
22. R. G. Edwards et al. [LHPC Collaboration], Phys. Rev.
Lett.96 (2006) 052001.
23. A.Ali Khan et al.,arXiv:hep-lat/0603028.
24. V. Bernard and U. G. Meissner, Phys. Lett. B 639, 278
(2006).