BARYON STRUCTURE FROM LATTICE QCD

1
BARYON STRUCTURE FROM LATTICE QCD
?. ??e???d???
?a?ep?st?µ?? ??p???

• C. Alexandrou
• University of Cyprus and Cyprus Institute

First European CLAS12 Workshop, Feb.25-28,
2009, Genoa, Italy
2
CLAS12 PROJECT

• Science Program
• Baryon form factors
• Generalized Parton Distributions
• Electroproduction at very small Q2
• Inclusive nucleon structure
• Semi-Inclusive DIS
• Properties of QCD from nuclear medium

3
LATTICE FERMIONS
DetD Difficult to simulate ?90s DetD1
quenched
valence

• A number of different discretization schemes have
  been developed to avoid doubling
• Wilson add a second derivative-type term ?
  breaks chiral symmetry for finite a
• Staggered distribute 4-component spinor on 4
  lattice sites ? still 4 times more species, take
  4th root, non-locality

Nielsen-Ninomiya no go theorem impossible to
have doubler-free, chirally symmetric, local,
translational invariant fermion lattice action
BUT
4
CHIRAL FERMIONS
Instead of a D such that ?5, D0 of the no-go
theorem, find a D such that ?5,D2a D ?5D

Ginsparg - Wilson relation

• Kaplan Construction of D in 5-dimensions ?
  Domain Wall fermions

Luescher 1998 realization of chiral
symmetry but NO no-go theorem!

• Neuberger Construction of D in 4-dimensions
  but with use of sign function ? Overlap
  fermions

Equivalent formulations of chiral fermions on
the lattice
5
REACHING THE CHIRAL REGIME
? Impressive progress in unquenched simulations
using various fermions discretization schemes

• With staggered, Clover improved and Twisted mass
  dynamical fermions results are emerging with pion
  mass of 300 MeV and in some cases lower

• PACS-CS using Clover NF21 fermions reported
  simulations with 160 MeV pions! But volume
  effects might be a problem

? Begin to address more complex observables e.g.
excited states and resonances, decays, finite
density density
6
DYNAMICAL SIMULATIONS

• Reaching the chiral regime
• Twisted mass ETMC (Cyprus, France, Germany,
  Italy, Netherlands, UK, Spain,
  Switzerland) automatic
  O(a2) but breaks isospin
• Clover QCDSF, UKQCD, PACS-CS, BMW (requires
  improvement of operators)
• Hybrid approach LHPC/Cyprus
• Domain wall RBC and UKQCD
• Overlap JLQCD

O(a2)
chiral fermions (expensive)
Most simulations are done using volumes larger
than 2 fm
Chiral extrapolation of lattice results are
becoming more reliable
7
SUMMARY OF DYNAMICAL SIMULATIONS
K. Jansen, Lattice 2008
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CHIRAL EXTRAPOLATIONS

• Controlled extrapolations in volume, lattice
  spacing and quark masses still needed for
  reaching the physical world - this may change in
  the next 2-3 years
• Quark mass dependence of observables teaches
  about QCD - can not be obtained from experiment
• Lattice systematics are checked using
  predictions from ?PT

Chiral perturbation theory in finite volume
Correct for volume effects depending on pion mass
and lattice volume
W. Detmold
Like in infinite volume
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NUCLEON MASS
r0mN
Good agreement for the nucleon mass
10
LIGHT HADRON MASSES
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INCLUDING THE STRANGE
r0mN
preliminary
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LIGHT HADRON MASSES

• NF21 Clover fermions stout smearing
• 3 lattice spacings 0.125 fm, 0.085 fm and 0.065
  fm
• volumes gt 2 fm
• lightest pion mass 190 MeV

S. Durr et al. Science 322 (2008) 1224
13
BARYON STRUCTURE

• Hybrid (mixed) action (LHPC/Cyprus),
• Twisted mass fermions (ETMC),
• Clover fermions (QCDSF, UKQCD, CPACS-CS, BMW,
  CERN),
• Domain wall fermions (RBC-UKQCD),
• Overlap fermions (JLQCD)

14
NUCLEON ELECTROMAGNETIC FORM FACTORS
Nucleon 3-point functions
Disconnected diagram omitted so far. Better
methods are being developed to allow their
computation e.g. dilution, lower eigenvalue
projection, one-end trick,
From connected diagram we calculate the isovector
Sachs form factors
GvE,M GpE,M GnE,M
PQCD
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EXPERIMENT
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Nucleon EM form factors
Cyprus/MIT Collaboration, C. A., G. Koutsou, J.
W. Negele, A. Tsapalis
QCDSF/UKQCD
Dirac FF
q2 - Q2
Results using domain wall valence on staggered
sea are from LHPC
17
CHIRAL EXTRAPOLATION
Heavy baryon effective theory with explicit ?
degrees of freedom
C. Alexandrou et al., PRD 71
18
Recent lattice results

• C. Alexandrou et al., PRD 74 (2006) 034508
  (ETMC) PoS LAT2008, arXiv0811.0724
• Ph. Hagler et al.,(LHPC) PRD 77 (2008) 094502
• M. Gockeler et al., (QCDSF) PRD 71 (2005)
  034508 PoS LAT2007,161, Pos LAT 2006
• H.-W. Lin et al., PRD 78 (2008) 014505
• Sh. Ohta and T. Yamazaki et al., (RBC-UKQCD) PoS
  LAT2008, arXiv0810.0045
• S. Sasaki and T. Yamazaki, PRD 78 (2008) 014510

19
RECENT RESULTS ON EM NUCLEON FORM FACTORS
JLab group is developing methods to go to higher
Q2. Results obtained in quenched approximation
for lowest mp 500 MeV
20
Dirac and Pauli r.m.s. radii
Twisted mass and domain wall fermions
21
Nucleon axial charge
Obtain axial nucleon form factors without
computational cost
Forward matrix element yields gAGA(0)

• accurately measured 1.2695(29)
• no-disconnected diagrams
• chiral PT

? benchmark for lattice calculations
22
RECENT RESULTS ON THE NUCLEON AXIAL CHARGE
QCDSF
Lattice volume (2.7 fm)3
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NUCLEON AXIAL FORM FACTORS
Within the fixed sink method for 3pt function we
can evaluate the nucleon matrix element of any
operator with no additional computational cost

• use axial vector current
  to obtain the axial form factors
• pseudoscalar density
  to obtain GpNN(q2)

24
GA AND GP
Results in the hybrid approach by the LPHC in
agreement with dynamical Wilson results
Unquenching effects large at small Q2 in line
with theoretical expectations that pion cloud
effects are dominant at low Q2
dipole fit to experimental results
pion pole dominance
Cyprus/MIT collaboration
dipole fit
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DISTRIBUTION AMPLITUDES
Exclusive processes at large Q2 can be
factorized into

• perturbative hard scattering amplitude (process
  dependent)

• nonperturbative wave functions describing the
  hadron's overlap with lowest Fock state (process
  independent)

experiment
Helicity amplitudes A1/2 and S1/2 can be
expressed in terms of G1(q2) and G2(q2)
lattice
Small volumes and pion mass higher than 400 MeV
V. M. Braun et al., arXiv0902.3087
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? ELECTROMAGNETIC FORM FACTORS
Pµ total momentum A linear combination of a1, a2,
c1, c2 gives GE0, GM1, GE2 and GM3
subdominant
C. A. et al., PRD79014507(2009), arXiv0901.3457 
Exponential fits
Dipole fits
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? ELECTRIC QUADRUPOLE FORM FACTOR
Also P. Moran et al. Adelaide, quenched results
Exponential fits
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N TO ? TRANSITION FORM FACTORS
Electromagnetic N to ? Write in term of Sachs
form factors
29
COMPARISON WITH EXPERIMENT
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QUADRUPOLE FORM FACTORS AND DEFORMATION
Q20.127 GeV2

• Quantities measured in the lab frame of the ?
  are the ratios

• Precise experimental data strongly suggest
  deformation of Nucleon/?
• First conformation of non-zero EMR and CMR in
  full QCD

C. N. Papanicolas, Eur. Phys. J. A18 (2003)
Large pion effects
31
CHIRAL EXTRAPOLATION
Just beginning to enter chiral regime
NLO chiral extrapolation on the ratios using
mp/?d2 , ?/?d. GM1 itself not given. V.
Pascalutsa and M. Vanderhaeghen, hep-ph/0508060
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AXIAL N TO ?
dipole
M. Procura, arXiv0803.4291 SSE to O(e3)
C5A(Q2,mp)a1a2mp2a3Q2loop
exponential
Again unquenching effects at small Q2 clearly
visible Pion-pole dominance heavier pion mass
required
C6(Q2) C5(Q2) c0/(m2Q2)
33
G??? AND G???
Curves refer to the quenched data

• Q2-dependence for Gp?? (Gp??) extracted from GA
  (C5A) using the Goldberger-Treiman relation
  deviates at low Q2
• Approximately linear Q2-dependence (small
  deviations seen at very low Q2 in the case of
  Gp?? need to be checked)
• Gp??/Gp?N 2C5A/GA as predicted by taking
  ratios of GTRs
• Gp??/Gp?N1.60(1)

GTR GpN?GAmN/fp
Gp?Na(1-bQ2/mp2)
Gp??1.6GpN?
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MOMENTS OF PARTON DISTRIBUTIONS
Parton distributions measured in deep inelastic
scattering
Forward matrix elements
Moments of parton distributions
3 types of operators
unpolarized moment
polarized moment
transversity
35
MOMENTS OF GENERALIZED PARTON DISTRIBUTIONS
Off diagonal matrix element Genelarized
parton distributions
GPD measured in Deep Virtual Compton Scattering
Generalized FF
GPDs
t0 (forward) yield moments of parton
distributions
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TRANSVERSE QUARK DENSITIES
as n increases slope of An0 decreases
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GENERALIZED FORM FACTORS A20, B20, C20
LHPC PRD 77, 094502(2008), 0705.4295
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CHIRAL EXTRAPOLATION
LHPC
ltxgtu-d
NLO gA,0--gt gA,lat, same with fp Self consistent
HB?PT
QCDSF, Lattice 2007 0710.1534
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ltXgtu-d AND ltXgt?u-?d
Finite size effect?
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SPIN STRUCTURE OF THE NUCLEON
Total spin of quark
A20(0)
Spin of the nucleon
QCDSF/UKQCD Dynamical Clover
For mp340 MeV Ju0.279(5) Jd-0.006(5)
Lud-0.007(13)
?Sud0.558(22)
In agreement with LPHC, Ph. Hägler et al.,
hep-lat/07054295
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EXCITED STATES
Several approaches

• Phase shift method M. Luscher
• Correlation matrix C. Michael M. Luscher S.
  Barak et al., PDR76, 074504 (2007)
  C. Gattringer,
  arXiv0802.2020 B. G. Lasscock et al., PRD 76
  (2007) 054510 C. Morningstar, Lattice
  2008.
• Maximum entropy methods P. Lepage et al., Nucl.
  Phys. Proc. Suppl. 106 (2002) 12
  C. Morningstar, Nucl. Phys. Proc. Suppl. 109A
  (2002) 185.
• ?2-method C. Alexandrou, E. Stiliaris, C.N.
  Papanicolas, PoS LAT2008, arXiv0810.3982
• Histogram method V. Bernard et al.,
  arXiv0806.4495

N-Roper transition matrix element H.-W. Lin et
al.
So far quenched, heavy pions
42
MATRIX OF CORRELATORS

• For a given NxN correlator matrix
  one defines the N
  principal correlators ?k(t,t0) as the eigenvalues
  of
• 
  where t0 is small
• The N principal effective masses tend (plateau)
  to the N lowest-lying stationary-state energies

C. Morningstar, Lattice 2008

• Crucial to use very good operators so noise does
  not swamp signal
• construct operators using smeared fields
• - link variable smearing
• quark field smearing
• spatially extended operators
• use large set of operators (variational
  coefficients)

43
?c spectrum
?2 - method
Analysis by C. Davies et al. using priors (P.
Lepage et al. hep-lat/0110175) 1.3169(1)
1.62(2) 1.98(22) Compare to
?2-analysis 1.3171(13) 1.608(9)
2.010(11)
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NUCLEON SPECTRUM
C. Alexandrou, E. Stiliaris, C.N. Papanicolas,
PoS LAT2008, arXiv0810.39
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CONCLUSIONS

• Lattice QCD is entering an era where it can
  make significant contributions in the
  interpretation of current experimental results.
• A valuable method for understanding hadronic
  phenomena

• Accurate results on coupling constants, form
  factors, moments of generalized parton
  distributions are becoming available close to the
  chiral regime (mp300 MeV) ? expected to below
  200 MeV in the next couple of years
• More complex observables are evaluated e.g
  excited states, polarizabilities, scattering
  lengths, resonances, finite density
• First attempts into Nuclear Physics e.g. nuclear
  force

• Computer technology and new algorithms will
  deliver 100s of Teraflop/s in the next five
  years

• Provide dynamical gauge configurations in the
  chiral regime
• Enable the accurate evaluation of more involved
  matrix elements

We expect a lot of interesting Physics
46
STRANGENESS

• Indirect charge symmetry chiral
  extrapolation D.Leinweber et al., PRL 94, 212001
  (2005) 97, 022001(2006) H.-W Lin and K. Orginos
  (mixed action)
• Direct disconnected contributions, still noisy
  R. Babich et al. , LAT2008

GsE
GsM-0.082(8)(25) GsE 0.00044(1)(130)
J. Zanotti, Lattice 2008
47
PION SECTOR
Applications to pion and nucleon mass and decay
constant e.g.
G. Colangelo, St. Dürr and Haefeli, 2005
Pion decay constant
NLO continuum ?PT
Applied by QCDSF to correct lattice data
r00.45(1)fm
48
??-SCATTERING LENGTHS

• More statistics
• mixed action ?PT formula

mpa02 -0.04330(42) A. Walker-Loud,
arXiv0706.3026
pp s-wave length a00 and a02
ETMC
Leutwyler (hep-ph/0612112)
49
RATIOS
50
RATIOS
51
Dirac and Pauli rms radii
Sh. Ohta and T. Yamazaki, PoS LAT2008
52
TRANSVERSE SPIN DENSITY
QCDSF Ph. Hägler et al. PRL 98, 222001 (2007)
q unpolarized
N unpolarized
u
N
N
d
53
MAGNETIC DIPOLE FORM FACTOR
54
TWISTED MASS DYNAMICAL FERMIONS
Consider two degenerate flavors of quarks Action
is
Twisted mass µ t3 acts in flavor space
Wilson Dirac operator
untwisted mass
Advantages - Automatic O(a) improvement (at
maximal twist) - Only
one parameter to tune (zero PCAC mass at smallest
µ) - No additional
operator improvement
Disadvantages - explicit chiral symmetry
breaking (like in all Wilson)
- explicit breaking of flavour symmetry
appears only at O(a2) in practice only affects
p0
Physics results reported for 270 MeV pions on
spatial size gt 2.1 fm with a lt 0.1fm
55
TWISTED MASS FORMULATION
Continuum fermionic action
The mass term can be written as
with atan-1(µ/m0),
M2m02µ2
At maximal twist ?p/2 and the mass is due to the
twisted mass term One parameter to tune just like
with Wilson fermions. We tune m0 to its critical
value by requiring the PCAC mass to vanish. This
is done at each ß value at the lowest
µ-value. This procedure is shown to preserve O(a)
improvement
56
DELTA MASS SPLITTING
Symmetry u d ? ? is degenerate with
?- and ? with ?0 Check for flavor breaking by
computing mass splitting between the two
degenerate pairs
ß3.8 L2.4 fm
Splitting consistent with zero in agreement with
theoretical expectations that isospin breaking
only large for neutral pions (16 on finest
lattice)
57
AXIAL NUCLEON FORM FACTORS
58
PION FORM FACTOR
S. Simula , Lattice07
59
BARYON SECTOR

• Volume effects small
• cutoff effects small
• Agreement with staggered fermion results
• Effect of dynamical strange quark small

60
NUCLEON MASS
HB?PT to lowest order Two fit parameters ?
m00.865(10), c1-1.224(17) GeV-1 ? s-term
66(3) MeV
statistical
Higher order yield results in agreement with the
lowest order
61
FIX LATTICE SPACING
Value of lattice spacing using the nucleon mass
at the physical point in agreement with that
extracted from the pion sector
non-trivial check of our lattice systematics
62
CONTINUUM LIMIT
Use the data at ß3.9 and 4.05 to estimate the
continuum limit and check consistency with ß3.8
Impressive agreement - cutoff effects non-visible
63
FITS TO CONTINUUM RESULTS
Lattice prediction of nucleon mass in agreement
with experiment
64
FITS TO ? MASS AT THE CONTINUUM LIMIT
Using r0 from the nucleon mass
65
GOLDBERGER-TREIMAN RELATIONS
Deviations from GTR
Improvement when using chiral fermions
66
RHO FORM FACTORS
lt?(p)jµ?(p)gt decomposed into three form
factors Gc(Q2), GM(Q2), GQ(Q2)
Related to the ? quadrupole moment GQ(0)m?2 Q?
Adelaide group Q? non-zero --gt rho-meson
deformed in agreement with wave function results
using 4pt functions
Non-relativistic approx.
?(y)2
One-end trick improves signal, application to
baryons is underway
G. Koutsou, Lattice07
67
HADRON-SHAPE
Density-density correlators evaluated exactly
using all-to-all propagators on dynamical two
degenerate flavors of Wilson fermions
For mesons we applied the one-end trick ?improves
statistical noise Rho clearly prolate
68
PION FORM FACTOR
69
COMPARISON WITH EXPERIMENT
Data, assuming only M1
p-cloud
Thanks V. Burkert
Analysis by Cole Smith to the CLAS p0 data, B.
Julia-Diaz, et al. PRC75 (2007)
Thanks L. Tiator
70
QUADRUPOLE FORM FACTOR GC2