n u c l e o n s tr u c tu r e o n th e l a ttic...
Transcript of n u c l e o n s tr u c tu r e o n th e l a ttic...
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