Πανεπιστήμιο Κύπρου Κ. Αλεξάνδρου BARYON STRUCTURE FROM LATTICE QCD C. Alexandrou University of Cyprus and Cyprus Institute First European CLAS12 Workshop,

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


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


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


SUMMARY OF DYNAMICAL SIMULATIONS

K. Jansen, Lattice 2008


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


NUCLEON MASS

Good agreement for the nucleon mass

r 0mN

No common scaling

r 0 fπ


LIGHT HADRON MASSES

INCLUDING THE STRANGE


(r0mπ)2

r 0mN


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


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


EXPERIMENT


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


CHIRAL EXTRAPOLATION

Heavy baryon effective theory with explicit Δ degrees of freedom

C. Alexandrou et al., PRD 71


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.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


Twisted mass and domain wall fermions


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


RBC/UKQCD, T. Yamazaki et al. PRL 100, 171602, 2008

Finite volume dependence

QCDSF

Lattice volume (2.7 fm)3


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


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


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


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


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


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


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)



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


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


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


< 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


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


LHPC: PRD 77, 094502(2008), 0705.4295



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


Finite size effect?

SPIN STRUCTURE OF THE NUCLEON


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


• 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


• 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


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, E. Stiliaris, C.N. Papanicolas, PoS LAT2008, arXiv:0810.39


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


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


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


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


RATIOS


RATIOS


Dirac and Pauli rms radii

C.A.

Sh. Ohta

H.-W. Lin

S. Sasaki

Sh. Ohta and T. Yamazaki, PoS LAT2008


TRANSVERSE SPIN DENSITY

Unpolarized

u

d

N

N

u

d

q unpolarized N unpolarized

QCDSF: Ph. Hägler et al. PRL 98, 222001 (2007)


MAGNETIC DIPOLE FORM FACTOR


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


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:


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 Hadron Electromagnetic form factors, ECT*, May 12-23 2008

PION FORM FACTOR

Comparison with experiment

S. Simula , Lattice07


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


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


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


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


FITS TO CONTINUUM RESULTS

Lattice prediction of nucleon mass in agreement with experiment


FITS TO Δ MASS AT THE CONTINUUM LIMIT

Using r0 from the nucleon mass


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 )


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


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


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


QUADRUPOLE FORM FACTOR GC2

Documents

Πανεπιστήμιο Κύπρου Κ. Αλεξάνδρου BARYON STRUCTURE FROM LATTICE QCD C. Alexandrou University of Cyprus and Cyprus Institute First European CLAS12 Workshop,