nucleon structure on the lattice

施羅斯Wolfram Schroers

國立 台灣 大學

QCDSF Collaboration

M. Göckeler, Ph. Hägler, R. Horsley,Y. Nakamura, M. Ohtani, D. Pleiter, P.E.L. Rakow,

G. Schierholz, H. Stüben, J. Zanotti

LHPC Collaboration

Ph. Hägler, J. Bratt, R.G. Edwards, M. Engelhardt,G.T. Fleming, B. Musch, J.W. Negele, K. Orginos,

A.V. Pochinsky, D.B. Renner, D.G. Richards

Outline

• Nucleon structure: phenomenology

• Lattice simulations & their challenges

• Achievements: Five key results

• Summary

• Outlook

Nucleon structure:Phenomenology

DVCS

!p ′ !p

γ∗ γ

e + p → e + p + γSignature:

Factorization ansatz

hard

soft

!p!p ′

γ∗ γ

(x − ξ)p+ (x + ξ)p+

x avg. long. mom.ξ long. mom. transfer

t = ∆2 = (p′ − p)2 virtuality

Forward limit

!p!p

γ∗ γ∗

xp+ xp+

⇒ Recover forward parton distributions

Local limit

!p ′ !p

γ∗

⇒ Recover form factors

QCD matrix element

!p ′ !p

ψ(−z−/2

)ψ

(z−/2

)

p+

∫dz−

2πeip+z−

〈p′|ψ(−z−/2

)γ5γ

+ψ(z−/2

)|p〉

= H(x, ξ, t)〈〈γ5γ+〉〉 − E(x, ξ, t)∆+

2m 〈〈γ5〉〉H(x, ξ, t) E(x, ξ, t)

Interpreting GPDs

!p ′ !p

ψ(−z−/2

)ψ

(z−/2

)

• Quark emitted and absorbed with l.m.f. (x+ξ) and (x-ξ)

• Quark/antiquark pair emitted with l.m.f. (ξ+x) and (ξ-x)

Collection of GPDs

〈p′|ψγ5γµψ|p〉 ⇒ H(x, ξ, t)& E(x, ξ, t)

〈p′|ψγµψ|p〉 ⇒ H(x, ξ, t)&E(x, ξ, t)

+ four more fermion GPDs for transversity

+ eight more gluon GPDs

Experimental signatures

• Deeply virtual-wide-angle Compton scattering

• Exclusive meson production

• Form factors, Deep-inelastic scattering

• Three-dim. hadron structure

• Angular momentum sum rule

Recent review: Phys.Rept. 388:41-277 (2003)

Lattice simulations & their challenges

Merits of lattice QCD

• Goal: Qualitative & quantitative results from first principles

⇒ Comparison of theory ⇔ experiment

⇒ Credibility for predictions

• Vary parameters, e.g. mq, Nc, Nf

• Test models of QCD ⇒ Insight into how QCD works

Regimes of quark masses

• Heavy quark regime:

confinement, flux tubes, adiabatic potential

• Light quark regime:

chiral symmetry breaking, instantons, chiral perturbation theory

Regimes of quark masses

• Heavy quark regime:

confinement, flux tubes, adiabatic potential

• Light quark regime:

chiral symmetry breaking, instantons, chiral perturbation theory

Major source ofuncertainty:

Quark masses!

• (Improved) Wilson fermions

• (Improved) staggered fermions

• Twisted mass Wilson fermions

• Ginsparg-Wilson fermions

• Domain-wall

• Overlap

Fermion discretizations

Practically very important

question!

• (Improved) Wilson fermions

• (Improved) staggered fermions

• Twisted mass Wilson fermions

• Ginsparg-Wilson fermions

• Domain-wall

• Overlap

Fermion discretizations

Cheap

(Improved) staggered fermions

Simple,

well

understood

(Improved) Wilson fermions

Chiral, O(a 2),but expensive

Ginsparg-Wilson fermions

Domain-wallOverlap

Twisted mass Wilson fermionsSmall m

PS

problematic

• (Improved) Wilson fermions

• (Improved) staggered fermions

• Twisted mass Wilson fermions

• Ginsparg-Wilson fermions

• Domain-wall

• Overlap

Fermion discretizations

Practically very important

question!

Cheap

(Improved) staggered fermions

Simple,

well

understood

(Improved) Wilson fermions

Chiral, O(a 2),but expensive

Ginsparg-Wilson fermions

Domain-wallOverlap

Twisted mass Wilson fermionsSmall m

PS

problematic

Improvements inalgorithms

Improvements incomputer hardware

• Light valence fermions possible today

• Light sea quarks remain major issue

• Hybrid calculations

• Full GW: Either full DWF or Overlap

• Full Wilson-type fermions (Clover/Twisted mass)

The challenge

Possible solutions:

Nucleon mass (QCDSF)

Extracting GPDs

• x-dependence: as Mellin-moments

• ξ-dependence: analytically

• t-dependence: via external momenta

• On the lattice: model-independent results

• Experimentally: difficult to extract functions of three variables ⇒ Combine lattice, models, and experiment

Achievements

Key results

• Five key results from lattice calculations:

• GPDs - Quark contribution to nucleon spin

• GPDs - Nucleon transverse structure

• Form factors: scaling @ large Q2

• N→Δ transition form factors

• Nucleon axial coupling gA : First quantitative result!

Quark spin contribution

1

2= Jq + Jg =

1

2Σ + Lq + Jg

Decomposition of nucleon spin:

1

2Jq

1

2Σ LqLq

Quark spin contribution

Phys.Rev.Lett. 92:042002 (2004)

LHPC: Hybrid calculations

• Hybrid approach: unitarity & square root? Lattice artifacts?

• Achievement: 5 quark masses, full QCD down to mπ=354 MeV

• Lattice sizes (2.5fm)3 and (3.5fm)3

LHPC hybrid action

arXiv:0705/4295

0.1 0.2 0.3 0.4 0.5 0.6mΠ2 !GeV2"

0.05

0.1

0.15

0.2

0.25

0.3Ju"d

0.1 0.2 0.3 0.4 0.5 0.6mΠ2 !GeV2"

0.05

0.1

0.15

0.2

0.25

0.3Ju"d

LHPC hybrid action

arXiv:0705/4295

0.2 0.4 0.6 0.8mΠ2 !GeV2"

0

0.1

0.2

0.3

0.4

contributionstonucleonspin

Lu"d

#$u"d#2

0.2 0.4 0.6 0.8mΠ2 !GeV2"

0

0.1

0.2

0.3

0.4

contributionstonucleonspin

LHPC hybrid action

0.2 0.4 0.6 0.8mΠ2 !GeV2"

"0.2

0

0.2

0.4contributionstonucleonspin

Lu

Ld

#$u#2

#$d#20.2 0.4 0.6 0.8

mΠ2 !GeV2""0.2

0

0.2

0.4contributionstonucleonspin

arXiv:0705/4295

Remarkable Features

• Cancellation of OAM for u+d quarks

• Cancellation between OAM and spin contribution for down quarks

• Qualitative features over entire mass range

Other publications

• Similar approach as ours:Phys.Rev. D62:114504 (2000) (N. Mathur et al)

• Alternative approach: Direct computation of

Phys.Rev. D65:094510 (2002) (V. Gadiyak et al)

But: see Phys.Rev. D66:017502 (2002) (W. Wilcox)

〈!p|yjJi(y)|!p〉

Ohter Publications

Hadron transverse structure

H(x, ξ = 0,−∆2⊥)

q(x, b⊥)

measures Fourier transform of

Phys.Rev. D62:071503 (2000) (M. Burckardt)For ξ≠0: Phys.Rev. D66:111501 (2002) (Ralston,Pire)

Eur.Phys.J. C25:223 (2002) (M. Diehl)

y

xp

• Generalized parton distribution at =0

⊥zδ

b⊥

x

f x b( , )⊥

1

0

xz

b⊥

Nucl.Phys.Proc.Suppl. 128:203-210 (2004)

Schematic in i.m.f.

Moments of GPDs

An0(−∆2⊥) ≡

∫d2b⊥ dx xn−1q(x, b⊥)ei

!b⊥· !∆⊥

Do the moments depend on n or not?

Does x-dep. factorize?

0 0.5 1 1.5 2 2.5 3 3.5!t !GeV2"

0.2

0.4

0.6

0.8

1

An0u!d

A10,A20,A30 mΠ#897 MeV

Phys.Rev.Lett. 93:112001(2004)

Also for lighter quarks

0 0.5 1 1.5 2 2.5 3 3.5!t !GeV2"

0.2

0.4

0.6

0.8

1

An0u!d

A10,A20,A30 mΠ#744 MeV

Phys.Rev.Lett. 93:112001(2004)

LHPC,arXiv:0705/4295

A30A20A10

m 353MeV, 283, u d

A30A20A10

m 353MeV, 283, u d

A20A10

m 356MeV, 203, u d

A20A10

m 356MeV, 203, u d

A30A20A10

m 496MeV, 203, u d

A30A20A10

m 496MeV, 203, u d

A30A20A10

m 595MeV, 203, u d

A30A20A10

m 595MeV, 203, u d

A30A20A10

m 682MeV, 203, u d

A30A20A10

m 682MeV, 203, u d

A30A20A10

m 758MeV, 203, u d

A30A20A10

m 758MeV, 203, u d

0.20.40.60.81

1.2

A01,A

02,A

03

0.20.40.60.81

1.2

A01,A

02,A

03

0.20.40.60.81

1.2

A01,A

02,A

03

0.20.40.60.81

1.2

A01,A

02,A

03

0.20.40.60.81

1.2

A01,A

02,A

03

0.20.40.60.81

1.2

A01,A

02,A

03

0.20.40.60.81

1.2

A01,A

02,A

03

0.20.40.60.81

1.2

A01,A

02,A

03

0.20.40.60.81

1.2A01,A

02,A

03

0.20.40.60.81

1.2A01,A

02,A

03

0.20.40.60.81

1.2

A01,A

02,A

03

0.20.40.60.81

1.2

A01,A

02,A

03

0 0.2 0.4 0.6 0.8 1 1.2t GeV2

0 0.2 0.4 0.6 0.8 1 1.2t GeV2

0 0.2 0.4 0.6 0.8 1 1.2t GeV2

0 0.2 0.4 0.6 0.8 1 1.2t GeV2

Form factors

tF2(t)F1(t)

∝ const. Naive quark counting rules

√tF2(t)F1(t)

∝ const. JLab spin transfer expt.

tF2(t)log2(t)F1(t)

∝ const. Phys.Rev.Lett. 91:092003 (2003)

dependencet = Q2

Form factors in Nature

Phys.Rev.Lett. 91:092003 (2003)

0 1 2 3 4-t / GeV2

0.1

0.2

0.3

0.4

0.5

Form

fact

or ra

tio

κ = 0.1560κ = 0.1570

Nucl.Phys.Proc.Suppl. 128:170-178 (2004) (Negele et al)

In the heavy pion world

Form factor scalingQCDSF Collaboration

0 1 2 3 4

Q2 [GeV2]

0

0.1

0.2

(Q2 /lo

g2 Q2 /Λ

2 ) F2(p

) / F 1(p

) [GeV

2 ]

β=5.25, κsea=0.13575β=5.29, κsea=0.13590β=5.40, κsea=0.13610

mPS≈600 MeV, a=0.070...0.084 fm

0 0.3 0.6 0.9 1.2Q2 (GeV2)

0

1

2

3Q

F2(I=

1) /

F 1(I=1)

(G

eV)

Expt: one σ bandmπ = 353 MeV, 3.5 fm3

mπ = 761 MeV, 2.5 fm3Preliminary

LHPC collaboration, in preparation

Hadron deformation

• How to measure?

• Quadrupole moment of ground state⇒ identical to zero for spin-1/2 system

• Excitation spectrum of the system⇒ exceedingly complicated, broad & overlapping resonances

• Radiation of emitted de-excitation radiation⇒ viable from Δ+(1232)

Transition form factors

Electromagnetic current (local operator):〈state|ψγµψ|state〉

Expand m.e. in terms of scalar functions(form factors or “generalized form factors”):

Transition form factors:〈∆σ(p′)|ψγµψ|n(p)〉 = Aσµ

1GM1(t) + Aσµ

2GE2(t) + Aσµ

3GC2(t)

〈n(p′)|ψγµψ|n(p)〉 = Pµ

1F1(t) + Pµ

2F2(t)F1(t) F2(t)

GM1(t) GE2(t) GC2(t)

Observables & lattice m.e.

• Matrix element

• Signal for deformation:spherical ⇒ M1deformed ⇒ M1, E2, C2

• Ratios:

〈∆σ(p′)|ψγµψ|n(p)〉 = Aσµ1

GM1(t) + Aσµ2

GE2(t) + Aσµ3

GC2(t)

REM = −GE2(t)

GM1(t), RSM = −

|∆|

2m∆

GC2(t)

GM1(t)

First results

• M1 transition form factor: nucl-th/0012046quark models predict M1 30% too small

• Phys.Rev.D66:094503(2002) ⇒ REM, RSM

• Phys.Rev.Lett. 86,2963(2001)

• Phys.Rev.Lett. 88:122001(2002)

• Eur.Phys.J.A18,141(2003)Eur.Phys.J.A17,349(2003)

Comparison to experiment

• Define

• Perform fit (similar to experiment)

G∗

M1(t) =1

3

1√

1 + t

(mN+m∆)2

GM1(t)

Ga(t) = Ga(0)(1 + αt) exp(−γt)GpE(t)

Quenched results

Phys.Rev.Lett. 94:021601 (2005)

The axial coupling gA

• Fundamental property of the nucleon

• Governs β-decay

• Quantitative measure of spont. χSB in hadronic physics

• Known to high accuracy experimentally from neutron β-decay

• Forward limit of nucleon axial form factor

Chiral expansions

• Phys.Rev. D70:074029 (2004) (Beane&Savage)

• Phys.Rev. D71:054510 (2005) (Detmold & Lin)

• Phys.Rev. D68:075009 (2003) (Hemmert et.al.)

• Phys.Rev. D66:054501 (2002) (Detmold et.al.)

• Phys.Rev. D74:094508 (2006) (QCDSF coll.)QCDSF paper

LHPC paper

LHPC: Hybrid calculations

• Hybrid approach: Asqtad & DWF

• Achievement: 5% acc. at mπ=354 MeV

• Lattice sizes (2.5fm)3 and (3.5fm)3

• Six constants: fπ, mΔ-mN, gNΔ, gA, gΔΔ, C

• First three: physical values, others are fit

• Total error from constr. parameters: <1%

0 0.2 0.4 0.6 0.8

m!

2 (GeV

2)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

gA

LHPC/MILCLHPC/SESAMRBCKQCDSF/UKQCDQCDSF/UKQCD (small V)Experiment

W. Schroers, EPJ A31:784 (2007)

See also:

Phys.Lett.B639:278

Phys.Rev.Lett. 96:052001 (2006)

QCDSF: Full Wilson

• Several lattice spacings, mπ, and V ⇒ better fits from var. (mπ,L)

• Excellent statistics, but mπ>594 MeV

• Different parameterization of χPT exp.

• Two seperate fit strategies attempted: one with mπ=500-600 MeV (Fit “A”) and one with mπ<700 MeV (Fit “B”)

Fit “A”

Phys.Rev. D74:094508 (2006)

0.0 0.1 0.2 0.3 0.4 0.5m 2 [GeV2]

0.8

1.0

1.2

1.4g A

=5.20=5.25=5.29=5.40

Fit “B”

0.0 0.2 0.4 0.6 0.8m 2 [GeV2]

0.8

1.0

1.2

1.4g A

=5.20=5.25=5.29=5.40

Phys.Rev. D74:094508 (2006)

Combined result

0.0 0.1 0.2 0.3 0.4 0.5m 2 [GeV2]

0.6

0.8

1.0

1.2

1.4g A

L=L= 1.91 fmL= 1.27 fmL= 0.95 fm

Phys.Rev. D74:094508 (2006)

Summary: gA

Experiment (neutron β decay) gA = 1.2695(29)

PRL 96:052001 (2006) gA = 1.226(84)

PR D74:094508 (2006) gA = 1.31(9)(7)

Summary & Outlook

Summary

• Five key achievements

• Major progress in lattice simulations

• Qualitative insight into nucleon structure

• Quantitative results slowly becoming available

• Progress benefits from chiEFT

Outlook

• QCDSF simulations reach down to mPS≈350 MeV (2006), currently running mPS≈250 MeV (2008)

• Hope to reach mPS≈200 MeV by the end of this decade

• LHPC focusses on full DWF, similar quark masses

• TWQCD: Full Overlap