Πανεπιστήμιο Κύπρου
Κ. Αλεξάνδρου
BARYON STRUCTURE FROM LATTICE QCD
C. Alexandrou
University of Cyprus and Cyprus Institute
First European CLAS12 Workshop, Feb.25-28, 2009, Genoa, Italy
C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009
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
C. Alexandrou,, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009
LATTICE FERMIONSa
μ
Uμ(n)=eigaAμ(n)
ψ(n)
€
SF = ψ (k)D(k,n)ψ (n)k,n
∑€
S = SG + SF
Det[D]: Difficult to simulate 90’s Det[D]=1 : quenched
⟨O⟩=∫ dUndψ ndψ n O[U,ψ ,ψ ]e−S
n∏
Z=
∫ dUndet[D]O[U,D−1]e−SG
n∏
Z
Integrate out
It is well known that naïve discretization of Dirac action
leads to 16 speciesS
F= d4x ψ(x) γμ∂μ +m⎡⎣ ⎤⎦ψ(x)∫ → ∂μψ(x) =
ψ(x+μ)-ψ(x-μ)2a
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…
valence
C. Alexandrou, University of Cyprus
CHIRAL FERMIONS
Instead of a D such that {γ5, D}=0 of the no-go theorem, find a D such that {γ5,D}=2a D γ5D
Ginsparg - Wilson relation
Luescher 1998: realization of chiral symmetry but NO no-go theorem!
Kaplan: Construction of D in 5-dimensions Domain Wall fermions
Left-handed fermionright-handed fermion
L5
Neuberger: Construction of D in 4-dimensions but with use of sign function Overlap fermions
D =1+ ε(Dw)
2, ε(Dw) =
Dw
Dw+ Dw
Equivalent formulations of chiral fermions on the lattice
But still expensive to simulate despite progress in algorithms
Compromise: - Use staggered sea (MILC) and domain wall valence (hybrid) - LHPC
- Twisted mass Wilson - ETMC
- Clover improved Wilson - QCDSF, UKQCD, PACS-CS, BMW
C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009
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 NF=2+1 fermions reported simulations with ~160 MeV pions! But volume effects might be a problem
RBC-UKQCD: Dynamical NF=2+1 domain wall fermions reaching mπ~300 MeV on ~3 fm volume
JLQCD: Dynamical overlap NF=2 and NF=2+1, ml ~ms/6, small volume (~1.8 fm)
Progress in algorithms for calculating D-1 as mπ physical value
Deflation methods: project out lowest eigenvalues Lüscher, Wilcox, Orginos/Stathopoulos
Chiral extrapolations more reliable as simulations use mπ in chiral regime e.g. using lattice results on fπ/mπ yield LECs to unprecedented accuracy
Begin to address more complex observables e.g. excited states and resonances, decays, finite density density
C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009
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
Chiral extrapolation of lattice results are becoming more reliable
chiral fermions (expensive)
O(a2)
Most simulations are done using volumes larger than 2 fm
C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009
SUMMARY OF DYNAMICAL SIMULATIONS
K. Jansen, Lattice 2008
C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009
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 χPT
Chiral perturbation theory in finite volume:
Correct for volume effects depending on pion mass and lattice volume
Like in infinite volume W. Detmold
C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009
NUCLEON MASS
Good agreement for the nucleon mass
r 0mN
No common scaling
r 0 fπ
C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009
LIGHT HADRON MASSES
INCLUDING THE STRANGE
C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009
(r0mπ)2
r 0mN
C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009
LIGHT HADRON MASSES
S. Durr et al. Science 322 (2008) 1224
•NF=2+1 Clover fermions + stout smearing
• 3 lattice spacings: 0.125 fm, 0.085 fm and 0.065 fm
• volumes > 2 fm
• lightest pion mass ~ 190 MeV
C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009
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)
Coupling constants: gA, gπNN, gπΝΔ, …
Form factors: GA(Q2), Gp(Q2), GE(Q2), GM(Q2), …., where Q2=-q2=(p’-p)2
Parton distribution functions: q(x), Δq(x),…
Generalized parton distribution functions: H(x,ξ,Q2), E(x,ξ,Q2),…
• Form factors, GPDS: three-point functions:
• Four-point functions: yield detailed info on quark distribution
only simple operators used up to now - need all-to-all propagators
• Masses: two-point functions: t2 t0
t1t2 t0
t1t2 t0
t’1
e.g. O for EM is
C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009
NUCLEON ELECTROMAGNETIC FORM FACTORS
connected
disconnected
g
valence quarks
sea quarks
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:
with GE and GM given in terms of F1 and F2
GE(q2 ) =F1(q
2 )−q2
(2mN )2
F2 (q2 )
GM (q2 ) =F1(q
2 ) + F2 (q2 )
PQCD:
GvE,M
= GpE,M
– GnE,M
Nucleon 3-point functions
C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009
EXPERIMENT
C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009
Nucleon EM form factors
Results using domain wall valence on staggered sea are from LHPC
Cyprus/MIT Collaboration, C. A., G. Koutsou, J. W. Negele, A. Tsapalis
q2 =- Q2
QCDSF/UKQCD
Fp =23
Fu −13
Fd
Fn =−13
Fu +23
Fd Dirac FF
C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009
CHIRAL EXTRAPOLATION
Heavy baryon effective theory with explicit Δ degrees of freedom
C. Alexandrou et al., PRD 71
C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009
Recent lattice results•C. Alexandrou et al., PRD 74 (2006) 034508; (ETMC) PoS LAT2008, arXiv:0811.0724
• 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, arXiv:0810.0045
• S. Sasaki and T. Yamazaki, PRD 78 (2008) 014510
valence sea NF a (fm) L (fm) lowest mπ
(GeV)
C. Alexandrou Wilson - 0 0.092 3 0.411
C. Alexandrou Wilson Wilson 2 0.077 1.8 0.380
C. Alexandrou (ETMC)
Twisted mass
Twisted mass
2 0.089 0.067
2.1, 2.7 0.268
Ph. Hagler (LHPC) DWF staggered 2+1 0.124 2.5, 3.5 0.293
M. Gockeler (QCDSF)
Clover - 0 0.1070.0770.058
1.71.81.9
0.5530.6170.505
M. Gockeler (QCDSF)
Clover Clover 2 0.080.065
2 0.350
H. -W. Lin DWF DWF 2 0.113 1.9 0.490
Sh. Ohta (RBC) DWF DWF 2+1 0.113 1.8, 2.7 0.330
S. Sasaki DWF - 0 0.15 1.8, 2.4, 3.6 0.390
RECENT RESULTS ON EM NUCLEON FORM FACTORS
C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009
C.A. et al. ETMCM. Lin et al. LHPC
JLab group is developing methods to go to higher Q2. Results obtained in quenched approximation for lowest mπ~ 500 MeV
Dirac and Pauli r.m.s. radii
C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009
Twisted mass and domain wall fermions
C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009
Nucleon axial charge
Forward matrix element yields gA=GA(0)
benchmark for lattice calculations
Obtain axial nucleon form factors without computational cost
HBχPT with Δ Hemmert et al.
Improved Wilson (QCDSF)
Pleiter, Lat07
M. Gockeler et al. PRD74 (2006) 094508
accurately measured: 1.2695(29)
no-disconnected diagrams
chiral PT
R. G. Edwards et al. PRL 96, 052001, 2006
Mixed action, LHPC
Savage & Beane, hep-ph/0404131
RECENT RESULTS ON THE NUCLEON AXIAL CHARGE
C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009
RBC/UKQCD, T. Yamazaki et al. PRL 100, 171602, 2008
Finite volume dependence
QCDSF
Lattice volume (2.7 fm)3
C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009
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
Aμ
a =ψγμγ5
τa
2ψ use axial vector current to obtain the axial form factors
pseudoscalar density to obtain GπNN(q2) Pa =ψiγ5
τa
2ψ
< N(
r′p , ′s ) |Aμ
3 |N(rp,s) >=i
mN2
EN ( ′p )EN (p)
⎛
⎝⎜
⎞
⎠⎟
1/2
u( ′p ) GA (Q2 )γμγ5 +
qμγ5
2mN
Gp(Q2 )
⎡
⎣⎢⎢
⎤
⎦⎥⎥
τ3
2u(p)
2m
q< N(
r′p , ′s ) |P3 |N(
rp,s) >=
mN2
EN ( ′p )EN (p)
⎛
⎝⎜
⎞
⎠⎟
1/2fπmπ
2GπNN (Q2 )
mπ2 +Q2
u( ′p ) iγ5
τ3
2u(p)
PCAC relates GA and Gp to GπNN:
G
A(Q2 ) -
Q2
4mN2
Gp(Q2 ) =
1
2mN
fπm
π2G
πNN(Q2 )
mπ2 + Q2
Generalized Goldberger-Treiman relation
Pion pole dominance:
1
2mN
Gp(Q2 ) ~
2GπNN
(Q2 )fπ
mπ2 + Q2
→ GπNN (Q2 )fπ =mNGA (Q
2 ) Goldberger-Treiman relation
C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009
GA AND GP
dipole fit
pion pole dominance:
dipole fit to experimental results
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
Cyprus/MIT collaboration
DISTRIBUTION AMPLITUDES
C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009
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)
First results on nucleon and N* [S11(1535)] distribution amplitudes are obtained by the QCDSF-UKQCD collaboration
Light cone sum rules derived for and making use of
dispersion relations one connects
distribution amplitudes for N* to the transition form factors for N* to N
Operator with the quantum numbers of the nucleon
Helicity amplitudes A1/2 and S1/2 can be expressed in terms of G1(q2) and G2(q2)
V. M. Braun et al., arXiv:0902.3087
experiment
lattice
Small volumes and pion mass higher than ~ 400 MeV
Δ ELECTROMAGNETIC FORM FACTORS
C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009
Pμ total momentum
A linear combination of a1, a2, c1, c2 gives GE0, GM1, GE2 and GM3
subdominantC. A. et al., PRD79:014507(2009), arXiv:0901.3457
Exponential fitsDipole fits
Δ ELECTRIC QUADRUPOLE FORM FACTOR
C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009
Exponential fitsAlso P. Moran et al. [Adelaide], quenched results
Quark transverse charge densities with definite light-cone helicity
Quark transverse charge densities in Δ+ polarized along the x-axis extracted from lattice data
ρΔ Τ3/2 ρΔ Τ1/2Cyprus-MIT/Mainz
C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009
N TO Δ TRANSITION FORM FACTORS
Electromagnetic N to Δ: Write in term of Sachs form factors:
< Δ(r′p , ′s ) |jμ |N(
rp,s) >=i
23
mΔmN
EΔ ( ′p )EN (p)
⎛
⎝⎜⎞
⎠⎟
1/2
uσ (r′p , ′s )Oσμ u(
rp,s)
Oσμ =GM1(q2 )K σμ
M1 +GE2 (q2 )K σμ
E2 +GC2 (q2 )K σμ
C2
Magnetic dipole: dominant
Electric quadrupole
Coloumb quadrupole
Axial vector N to Δ: four additional form factors, C3A(Q2), C4
A(Q2), C5A(Q2), C6
A(Q2)
Dominant: C5A GA
C6A GpPCAC relates C5
A and C6A to GπΝΔ :
C
5A (Q2 )−
Q2
mN2
C6A (Q2 ) =
12mN
fπmπ2GπNΔ (Q
2 )
mπ2 +Q2
Generalized non-diagonal Goldberger-Treiman relation
Pion pole dominance:
1
mN
C6A (Q2 ) :
1
2
G πNΔ (Q2 )fπ
mπ2 +Q2
→ GπNΔ (Q2 )fπ =2mNC5
A (Q2 )Non-diagonal Goldberger-Treiman relation
C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2008
COMPARISON WITH EXPERIMENT
GM1
Thanks L. Tiatorπ-cloud?
C. Alexandrou University of Cyprus Hadron Physics on the Lattice, Milos, Sept. 2007
QUADRUPOLE FORM FACTORS AND DEFORMATION
Q2=0.127 GeV2
Large pion effects
Quantities measured in the lab frame of the Δ are the ratios:
REM
(EMR) =−GE2 (Q
2 )GM1(Q
2 )
RSM (CMR) =−|rq|
2mΔ
GC2 (Q2 )
GM1(Q2 )
Precise experimental data strongly suggest deformation of Nucleon/Δ
First conformation of non-zero EMR and CMR in full QCD
C. N. Papanicolas, Eur. Phys. J. A18 (2003)
CHIRAL EXTRAPOLATION
C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009
NLO chiral extrapolation on the ratios using mπ/Μ~δ2 , Δ/Μ~δ. GM1 itself not given.
V. Pascalutsa and M. Vanderhaeghen, hep-ph/0508060
Just beginning to enter chiral regime
C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009
AXIAL N TO Δ
M. Procura, arXiv:0803.4291
SSE to O(ε3):
C5A(Q2,mπ)=a1+a2mπ
2+a3Q2+loop
C6A
mN2
=1
mπ2 + Q2
a1
- a4m
π2 - a
3Q2 + loop(m
π)⎡⎣ ⎤⎦
Again unquenching effects at small Q2 clearly visible
Pion-pole dominance: heavier pion mass required
C6(Q2 )
mN2
=C5 (Q
2 )mπ
2 +Q2
C6(Q2)= C5(Q2) c0/(m2+Q2)
dipole
exponential
GΠΝΝ AND GΠΝΔ
C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009
GπΝN=a(1-bQ2/mπ2)
GTR: GπNΝ=GAmN/fπ
GπΝΔ=1.6GπNΝ
Q2-dependence for GπΝΝ (GπΝΔ) extracted from GA (C5
A) using the Goldberger-Treiman relation deviates at low Q2
Approximately linear Q2-dependence (small deviations seen at very low Q2 in the case of GπΝΔ need to be checked)
GπΝΔ/GπΝN = 2C5A/GA as predicted by taking
ratios of GTRs
GπΝΔ/GπΝN=1.60(1)
Curves refer to the quenched data
MOMENTS OF PARTON DISTRIBUTIONS
C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009
< N(p, s) | O
q
μ1 ...μn{ } |N(p,s) >~ dx∫ xn-1q(x)
Forward matrix elements:
< xn >q= dx xn q(x) + (-1)n+1q(x)⎡⎣ ⎤⎦-1
1
∫ q=q↓ +q↑
< xn >Δq= dx xn
-1
1
∫ Δq(x) + (-1)nΔq(x)⎡⎣ ⎤⎦ Δq=q↓ -q↑
< xn >δq= dx xn
-1
1
∫ δq(x) + (-1)n+1δq(x)⎡⎣ ⎤⎦ δq=qT
+q⊥
unpolarized moment
polarized moment
transversity
Parton distributions measured in deep inelastic scattering
Oq
μ1 ...μn =q γμ1 itDμ2 ...i
tDμn}⎡
⎣⎤⎦q
O5qμμ1 ...μn =q γ5γ
μ itDμ1 ...i
tDμn}⎡
⎣⎤⎦q
OTqμνμ1 ...μn =q γ5σ
μν itDμ2 ...i
tDμn}⎡
⎣⎤⎦q
tD=
12(rD−
sD)
3 types of operators:Moments of parton distributions
MOMENTS OF GENERALIZED PARTON DISTRIBUTIONS
C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009
Off diagonal matrix element Genelarized parton distributions
GPD measured in Deep Virtual Compton Scattering
< N(p', s') | Oq
μ1 ...μn{ } |N(p,s) >=u( ′p , ′s ) Aniq (t)K ni
A +Bniq (t)K ni
B{ } +δn,evenCniq (t)K n
C
i=0even
n-1
∑⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥
u(p,s)
Generalized FF
Hn (ξ, t) ≡ dx-1
1
∫ xn-1 H(x, ξ, t)
En(ξ, t) ≡ dx-1
1
∫ xn-1 E(x, ξ, t)
Hn(0, t) =An0 (t) A10 (t) =F1(t)
En(0, t) =Bn0 (t) B10 (t) =F2 (t)
GPDs
P =12
p+ ′p( ) , q=(x+ ξ)p, ′q =(x−ξ)p
t=−Q2 =Δ2
t=0 (forward) yield moments of parton distributions
• Momentum sum rule:
• Spin sum rule
TRANSVERSE QUARK DENSITIES
C. Alexandrou, University of Cyprus Hadron Electromagnetic form factors, ECT*, May 12-23 2008
Fourier transform of form factors/GPDs gives probability
dx xn-1 -1
1
∫ q(x,rb⊥ ) =
d2Δ⊥
(2π )2∫ e-irb⊥ .
rΔ⊥ An0
q (-rΔ⊥
2 )
q(x→ 1,rb⊥ ) ∝δ 2(b⊥ )
Transverse quark distributions:
M. Burkardt, PRD62 (2000)
q(x,
rb⊥
2 ) =d2Δ⊥
(2π )2∫ e-irb⊥ .
rΔ⊥ H(x,ξ =0, -Δ⊥
2 )
as n increases slope of An0 decreases
GENERALIZED FORM FACTORS A20, B20, C20
C. Alexandrou,, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009
LHPC: PRD 77, 094502(2008), 0705.4295
CHIRAL EXTRAPOLATION
C. Alexandrou,, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009
NLO: gA,0--> gA,lat, same with fπ
Self consistent HBχPT
<x>u-d
LHPC
QCDSF, Lattice 2007: 0710.1534
<X>u-d AND <X>Δu-Δd
C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009
Finite size effect?
SPIN STRUCTURE OF THE NUCLEON
C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009
J
q=
1
2< x >
q+B
20q (0)( ) L q =J q -
12ΔΣq ΔΣ =%A10
q (0)
QCDSF/UKQCD Dynamical Clover
For mπ~340 MeV: Ju=0.279(5) Jd=-0.006(5)
Lu+d=-0.007(13) ΔΣu+d=0.558(22)
In agreement with LPHC, Ph. Hägler et al., hep-lat/07054295
Total spin of quark:
Spin of the nucleon:
1
2=Lu+d +
12ΔΣu+d + J g
A20(0)
EXCITED STATES
C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009
• Phase shift method: M. Luscher
•Correlation matrix: C. Michael; M. Luscher; S. Barak et al., PDR76, 074504 (2007); C. Gattringer, arXiv:0802.2020; 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.
• χ2-method: C. Alexandrou, E. Stiliaris, C.N. Papanicolas, PoS LAT2008, arXiv:0810.3982
• Histogram method: V. Bernard et al., arXiv:0806.4495
Several approaches:
N-Roper transition matrix element: H.-W. Lin et al.
So far quenched, heavy pions
MATRIX OF CORRELATORS
C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009
• For a given NxN correlator matrix one defines the N principal correlators λk(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)
ηc spectrum
C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009
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 χ2-analysis:
1.3171(13) 1.608(9) 2.010(11)
χ2 - method
NUCLEON SPECTRUM
C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009
C. Alexandrou, E. Stiliaris, C.N. Papanicolas, PoS LAT2008, arXiv:0810.39
C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009
CONCLUSIONS
Accurate results on coupling constants, form factors, moments of generalized parton distributions are becoming available close to the chiral regime (mπ~300 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
• 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
• Computer technology and new algorithms will deliver 100´s 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
STRANGENESS
C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009
Gs E
• 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
GsM=-0.082(8)(25)
GsE= 0.00044(1)(130)
J. Zanotti, Lattice 2008
C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009
PION SECTOR
Applications to pion and nucleon mass and decay constant e.g.
mπ −mπ (L)mπ
=-x2λ
I mπ
(2) (λ) +xI mπ
(4) (λ)⎡⎣
⎤⎦
|rn|≠0∑
fπ
- fπ(L)
fπ
= -x
λI fπ
(2) (λ) +xI fπ
(4) (λ)⎡⎣
⎤⎦
|rn|≠0∑
x=m0
2
(4πfπ0 )2
, λ =mπ |rn|L
G. Colangelo, St. Dürr and Haefeli, 2005
r0=0.45(1)fm
NLO continuum χPT
Pion decay constant
Applied by QCDSF to correct lattice data
C. Alexandrou, University of Cyprus
ΠΠ-SCATTERING LENGTHS
mπa02 =-0.04330(42)
A. Walker-Loud, arXiv:0706.3026ππ s-wave length a0
0 and a02
Leutwyler (hep-ph/0612112)
ETMC
C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009
RATIOS
C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009
RATIOS
C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009
Dirac and Pauli rms radii
C.A.
Sh. Ohta
H.-W. Lin
S. Sasaki
Sh. Ohta and T. Yamazaki, PoS LAT2008
C. Alexandrou, University of Cyprus
TRANSVERSE SPIN DENSITY
Unpolarized
u
d
N
N
u
d
q unpolarized N unpolarized
QCDSF: Ph. Hägler et al. PRL 98, 222001 (2007)
C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009
MAGNETIC DIPOLE FORM FACTOR
C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009
TWISTED MASS DYNAMICAL FERMIONS
Consider two degenerate flavors of quarks: Action is
S
F=a4 χ(x) Dw[U] +m0 + iμγ5τ3( )
x∑ χ(x)
Wilson Dirac operator untwisted mass
Twisted mass μ τ3 acts in flavor space
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 π0
Physics results reported for ~270 MeV pions on spatial size > 2.1 fm with a < 0.1fm
C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009
TWISTED MASS FORMULATION
S
F= d4x χ(x) γμDμ +m0 + iμγ5τ3( )∫ χ(x)
The mass term can be written as: with α=tan-1(μ/m0), M2=m02+μ2
m0+ iμγ5τ
3 =Meiαγ5τ3
ψ(x) =eiωγ5τ3 /2χ(x), ψ(x) =χ(x)eiωγ5τ
3 /2
Perform axial transformation to the physical basis
The mass term transforms as and if α=ω then two degenerate flavors we recover the standard action
Mei(α−ω)γ5τ3
S
F= d4xψ(x) γμDμ +M( )∫ ψ(x)
At maximal twist ω=π/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 β value at the lowest μ-value.
This procedure is shown to preserve O(a) improvement
Continuum fermionic action:
C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009
DELTA MASS SPLITTING
Symmetry: u d Δ++ is degenerate with Δ- and Δ+ with Δ0
Check for flavor breaking by computing mass splitting between the two degenerate pairs
Splitting consistent with zero in agreement with theoretical expectations that isospin breaking only large for neutral pions (~16% on finest lattice)
* β=3.8 L=2.4 fm
AXIAL NUCLEON FORM FACTORS
C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009
C. Alexandrou, University of Cyprus Hadron Electromagnetic form factors, ECT*, May 12-23 2008
PION FORM FACTOR
Comparison with experiment
S. Simula , Lattice07
C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009
BARYON SECTOR
- Volume effects: small
- cutoff effects: small
- Agreement with staggered fermion results
- Effect of dynamical strange quark small
Use continuum chiral PT to extrapolate to physical point
C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009
NUCLEON MASS β=3.9 β=3.9, 4.05
HBχPT to lowest order: Two fit parameters m0=0.865(10), c1=-1.224(17) GeV-1
σ-term =66(3) MeV
Higher order yield results in agreement with the lowest order
statistical
C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009
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
C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009
CONTINUUM LIMIT
Use the data at β=3.9 and 4.05 to estimate the continuum limit and check consistency with β=3.8
Impressive agreement - cutoff effects non-visible
C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009
FITS TO CONTINUUM RESULTS
Lattice prediction of nucleon mass in agreement with experiment
C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009
FITS TO Δ MASS AT THE CONTINUUM LIMIT
Using r0 from the nucleon mass
C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009
GOLDBERGER-TREIMAN RELATIONS
Deviations from GTR
Improvement when using chiral fermions
1
mN
C6A (Q2 ) :
1
2
G πNΔ (Q2 )fπ
mπ2 +Q2
→ GπNΔ (Q2 )fπ =2mNC5
A (Q2 )
1
2mN
Gp(Q2 ) :
2G πN (Q2 )fπ
mπ2 +Q2
→ GπN (Q2 )fπ =mNGA (Q
2 )
C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009
RHO FORM FACTORS
<ρ(p’)|jμ|ρ(p)> decomposed into three form factors Gc(Q2), GM(Q2), GQ(Q2)
Related to the ρ quadrupole moment GQ(0)=mρ
2 Qρ
Adelaide group: Qρ non-zero --> rho-meson deformed in agreement with wave function results using 4pt functions
Non-relativistic approx.
|Ψ(y)|2
One-end trick improves signal, application to baryons is underway
Decreasing quark mass
G. Koutsou, Lattice07
C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009
HADRON-SHAPE
For mesons we applied the one-end trick improves statistical noise
Rho clearly prolate
Density-density correlators evaluated exactly using all-to-all propagators on dynamical two degenerate flavors of Wilson fermions
mπ=380 MeVmπ = 680 MeV
For Δ we have results with dynamical quarks for the first time.
More noisy, small deformation in agreement with an oblate shape
X 0.1
C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009
PION FORM FACTOR
Set initial and final pion momentum to zero --> no disadvantage in applying the one-end trick
G. Koutsou, Lattice07
q-q
4-point function: first application of the one-end trick (C. Michael):
S. Simula, ETM Collaboration
one-trick for 3pt function
VMD :1
1−q2 / mρ2
Wilson
COMPARISON WITH EXPERIMENT
C. Alexandrou, University of Cyprus Hadron Electromagnetic form factors,
?
Thanks V. Burkert
Thanks L. Tiator
Data, assuming only M1
Analysis by Cole Smith to the CLAS π0 data, B. Julia-Diaz,
et al. PRC75 (2007)
π-cloud
C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009
QUADRUPOLE FORM FACTOR GC2