1. Low-energy precision observables and the role of lattice QCD
Hartmut Wittig
PRISMA Cluster of Excellence, Institute for Nuclear Physics and Helmholtz Institute Mainz
PAVI 14 – From Parity Violation to Hadron Structure,
Skaneateles, 17 July 2014

2. Low-Energy QCD and Standard Model tests
Structural properties of the nucleon:
Form factors, structure functions, GPDs
Contributions from gluons and the quark sea
Precision tests of the Standard Model at low energies:
QCD corrections to weak decay amplitudes
Hadronic contributions to the muon (𝑔−2)
Weak charge of the proton
[Courtesy of D. Becker]

3. Beyond Perturbation Theory: Lattice QCD
Non-perturbative treatment; regularised Euclidean functional integrals
lattice spacing: 𝑎, 𝑥 𝜇 = 𝑛 𝜇 𝑎, 𝑎 −1 ∼ Λ UV
finite volume: 𝐿 3 ⋅𝑇
Stochastic evaluation of 〈Ω〉 via Markov process
Simulation algorithm: Hybrid Monte Carlo
[Duane et al., 1987]
Strong growth of numerical cost near physical 𝑚 𝑢 , 𝑚 𝑑
Pion mass, i.e. lightest mass in the pseudoscalar channel:
≈500 MeV
(2001)
≈130−200 MeV
(2014) ⟶

4. Systematic Effects Lattice artefacts: Finite volume effects:
⟶ extrapolate to continuum limit from 𝑎≈0.05−0.12 fm
Finite volume effects:
Empirically: 𝑚 𝜋 𝐿 ≳4 sufficient for many purposes
Unphysical quark masses:
Chiral extrapolation to physical values of 𝑚 𝑢 , 𝑚 𝑑 becomes obsolete
Inefficient sampling of SU(3) group manifold:
Simulations trapped in topological sectors as 𝑎→0
Use open boundary conditions in time direction
[Lüscher & Schaefer, 2012]

5. Outline Status report: nucleon form factors and charge radii
Quark-disconnected diagrams
Numerical techniques
Strange form factors
Further applications
Scalar form factor of the pion
Disconnected contributions to 𝒂 𝝁 𝐇𝐕𝐏
Conclusions

6. I. Nucleon Form Factors and Charge Radii

7. Nucleon form factors and charge radii
Lattice simulations:
underestimate nucleon charge radii, axial charge 𝑔 𝐴
overestimate moments of PDFs: 𝑥 𝑢−𝑑
Systematic effects not fully controlled
Lattice artefacts
Chiral extrapolation to physical pion mass
Finite-volume effects
“Contamination” from excited states
Quark-disconnected diagrams ignored
[M. Lattice 2014]
Dirac and Pauli charge radii determined from “plateau method”:
Systematic effects not fully controlled
Lattice artefacts
Chiral extrapolation to physical pion mass
Finite-volume effects
“Contamination” from excited states
Quark-disconnected diagrams ignored

8. Statistical fluctuations in baryon correlators
Noise-to-signal ratio increases exponentially:
Nucleon at rest: 𝑅 NS 𝑥 0 ∼ e 𝑚 N − 𝑚 𝜋 𝑥 0
Pion at 𝑝 ≠0:
𝑅 NS 𝑥 0 ∼ e 𝑚 𝜋 2 + 𝑝 2 − 𝑚 𝜋 𝑥 0
Excited-state contributions die out slowly
Ground state dominates for 𝑎≳0.5 fm

9. Correlator ratios
Extract nucleon hadronic matrix elements from ratios of three- and two-point correlation functions, e.g.
Statistical fluctuations impose 𝑡 𝑠 ≲1.3 fm
Domination of the ground state doubtful

10. Methods for determining form factors
Plateau method: 𝑅 V 𝑞 ,𝑡, 𝑡 𝑠 = 𝐺 E,M ( 𝑄 2 )+𝑂 𝑒 −Δ𝑡 , 𝑒 − Δ ′ 𝑡 𝑠 −𝑡
Summed insertions:
[Maiani et al. 1987, Güsken et al. 1989, Bulava et al., Capitani et al. 2012]
𝑆 V 𝑡 𝑠 ≡ 𝑡=1 𝑡 𝑠 −1 𝑅 V 𝑞 ,𝑡, 𝑡 𝑠 = 𝐾 V + 𝑡 𝑠 𝐺 E,M ( 𝑄 2 )+𝑂 𝑒 −Δ 𝑡 𝑠 , 𝑒 − Δ ′ 𝑡 𝑠
Excited state contributions more strongly suppressed
Determine 𝐺 E,M ( 𝑄 2 ) from linear slope of summed ratio
Two-state fits: 𝑅 V 𝑞 ,𝑡, 𝑡 𝑠 = 𝐺 E,M 𝑄 2 + 𝑐 E,M (1) 𝑒 −Δ𝑡 + 𝑐 E,M (2) 𝑒 − Δ ′ ( 𝑡 𝑠 −𝑡)
Stabilise fits by fixing the energy gaps Δ and Δ ′

11. Methods for determining form factors
𝑅 V 𝑞 ,𝑡, 𝑡 𝑠 = 𝐺 E,M 𝑄 2 + 𝑐 E,M (1) 𝑒 −Δ𝑡 + 𝑐 E,M (2) 𝑒 − Δ ′ ( 𝑡 𝑠 −𝑡)
Plateau method: 𝑐 E,M (1) = 𝑐 E,M (2) =0
Summation method:
𝑆 𝑉 ( 𝑞 , 𝑡 𝑠 )= 𝐾 V + 𝑡 𝑠 𝐺 E,M ( 𝑄 2 )
[Jäger, Rae et al., arXiv: , and in prep.]

12. Methods for determining form factors
𝑅 V 𝑞 ,𝑡, 𝑡 𝑠 = 𝐺 E,M 𝑄 2 + 𝑐 E,M (1) 𝑒 −Δ𝑡 + 𝑐 E,M (2) 𝑒 − Δ ′ ( 𝑡 𝑠 −𝑡)
Plateau method: 𝑐 E,M (1) = 𝑐 E,M (2) =0
Chiral behaviour; comparison with LPHc
[Brambilla et al., ]
Summation method:
𝑆 𝑉 ( 𝑞 , 𝑡 𝑠 )= 𝐾 V + 𝑡 𝑠 𝐺 E,M ( 𝑄 2 )
[Jäger, Rae et al., arXiv: , and in prep.]

13. Methods for determining form factors
Chiral behaviour of 𝐺 𝐸 at 𝑄 2 =0.1 GeV:
[Jäger, Rae et al., arXiv: , and in prep.]

14. Summary: Nucleon electromagnetic form factors
Plateau method: 𝑡 𝑠 ≲1.2 fm not sufficient to rule out bias from excited state contributions
Agreement with phenomenology improved by
Detailed investigation of chiral behaviour and fits in progress
Removing excited state “contamination”
Using near-physical pion masses
Small finite-volume effects for 𝑚 𝜋 𝐿≳4
[Green et al., arXiv: ]

15. II. Quark-disconnected
Diagrams

16. Strangeness in the nucleon
Probe sea quark contributions to nucleon properties
Nucleon mass: 𝜋𝒩 𝜎-term
Strangeness form factors: 𝐺 𝐸 𝑠 𝑄 2 , 𝐺 𝑀 𝑠 ( 𝑄 2 )
Strangeness contribution to the nucleon spin:
1 2 = 1 2 ΔΣ+ 𝐿 𝑞 +Δ𝐺, ΔΣ=Δ𝑢+Δ𝑑+Δ𝑠+…
Contributions given entirely by quark-disconnected diagrams

17. Quark propagators in lattice QCD
Diagram includes
𝑥 Tr 𝛾 𝜇 𝑆(𝑥,𝑥)
Quark propagator:
𝑆 𝑥,𝑦 = 𝐷 −1 𝑥,𝑦 , 𝐷: lattice Dirac operator
Solve linear system: 𝐷𝜙=𝜂 ⟹ 𝜙 𝑥 = 𝐷 −1 (𝑥,𝑦)𝜂(𝑦)
Point source: 𝜂 𝑦 = 𝛿 𝑦0 ⟹ 𝜙 𝑥 = 𝐷 −1 𝑥,0 ≡𝑆(𝑥,0)
“point-to-all” propagator
Point source technique yields Tr[ 𝛾 𝜇 𝑆 0,0 ], i.e. only a single term
Must perform 𝐿 𝑎 3 inversions to obtain full contribution

18. Stochastic “all-to-all” propagators
Stochastic sources: 𝜂 𝑟 𝑥 ∈𝑈 1 , 𝑟=1,…, 𝑁 𝑟 , 𝑁 𝑟 =𝑂(10)
𝑁 𝑟 ⟶∞: ≪ 𝜂 𝑟 † 𝑥 𝜂 𝑟 ( 𝑦 )≫= 𝛿 𝑥 𝑦
(stochastic average)
Solve linear system: 𝐷𝜙=𝜂 ⟹ 𝜙 (𝑟) 𝑥 = 𝐷 −1 𝑥,𝑦 𝜂 (𝑟) ( 𝑦 )
Further refinement: hopping parameter expansion
[Bali, Collins, Schäfer 2010]
[Gülpers, von Hippel, H.W. 2013]
Stochastic noise parametrically suppressed
Method introduces additional (i.e. stochastic) noise
𝑥 Tr≪ 𝜂 𝑟 † 𝑥 𝛾 𝜇 𝜙 (𝑟) 𝑥 ≫ ≃ 𝑥 𝛿 𝑥 𝑦 Tr 𝛾 𝜇 𝑆 𝑥,𝑦
Stochastic average yields:

19. Strange form factors Doi et al., Phys Rev D80 (2009) 094503
Two-flavour QCD; 𝑂 𝑎 improved Wilson quarks; ⋅32
Single lattice spacing: 𝑎=0.121 fm
𝑚 𝜋 =600−840 MeV, 𝑚 𝜋 min 𝐿=5.9, 𝐿≈2 fm
𝑁 cfg =800 gauge configurations; 𝑁 src =64−82 point sources
Stochastic 𝑍(4) sources for disconnected part; 𝑁 r =600−800

20. Strange form factors Doi et al., Phys Rev D80 (2009) 094503
Fit 𝑄 2 -dependence to monopole / dipole form
Use summation method to determine form factors
Results:
𝐺 M 𝑠 0 =−0.017(25)(07)
𝑄 2 =0.1 GeV: 𝐺 M 𝑠 Q 2 =− , 𝐺 E 𝑠 Q 2 =0.0022(19)
Systematic uncertainties estimated from
Monopole/dipole ansatz for 𝑄 2 -dependence
Different fit formulae for chiral extrapolation
Estimate residual excited state contributions

21. Strange form factors Babich et al., Phys Rev D85 (2012) 054510
Two-flavour QCD; 𝑂 𝑎 improved Wilson fermions; ⋅64
Anisotropic lattice: 𝑎 𝑠 =0.108 fm, 𝑎 𝑠 𝑎 𝑡 ≈3
Single pion mass 𝑚 𝜋 ≈416 MeV, 𝑚 𝜋 𝐿=5.9, 𝐿≈2.6 fm, 𝑇≈2.3 fm
𝑁 cfg =863 gauge configurations
SU(3) unitary noise + “dilution” to reduce stochastic noise; 𝑁 r =864

22. Strange form factors Babich et al., Phys Rev D85 (2012) 054510
𝐺 𝐸 𝑠 , 𝐺 M 𝑠 compatible with zero at the permille / percent level (statistical error)

23. Summary: Strange form factors
Calculation far more demanding compared to electromagnetic form factors, due to disconnected diagrams
Many technical improvements
Gauge noise could still be large, even if stochastic noise is under control
Stochastic sources + “dilution”
Hopping parameter expansions
No significant deviation from zero observed so far
Compute disconnected diagrams on GPUs

24. III. Further Applications

25. Scalar form factor and radius of the pion
Definition; charge radius:
𝑞 2 = 𝑝 𝑓 − 𝑝 𝑖 2 =− 𝑄 2
𝐹 𝑠 0 ≡ 𝜎 𝜋
(pion 𝜎-term)
Chiral expansion:
ℓ 4 =4.783±0.097⟹ 𝑟 2 𝑠 =0.645±0.017 f m 2
Phenomenological determination from 𝜋𝜋-scattering
[Colangelo, Gasser & Leutwyler, Nucl Phys B603 (2001) 125]
𝑟 2 𝑠 =0.61±0.04 f m 2

26. Scalar form factor and radius of the pion
𝑂 𝑎 – improved Wilson fermions 𝑎=0.063 fm, 𝑚 𝜋 =280−650 MeV
Disconnected contribution evaluated using stochastic sources and HPE
[Gülpers, von Hippel, H.W., PRD89 (2014) ]
Fit to NLO:
𝑟 2 s =0.635± stat f m 2
Inclusion of disconnected part crucial for agreement with 𝜋𝜋-scattering

27. Hadronic vacuum polarisation on the Lattice
Lattice approach: evaluate convolution integral over Euclidean momenta
Time-momentum representation:
Method yields Π 𝑄 2 ≡Π 𝑄 2 −Π(0) without extrapolation to 𝑄 2 =0
Must determine vector correlator 𝐺 𝑥 0 for 𝑥 0 →∞
Include quark-disconnected contribution

28. Hadronic vacuum polarisation on the Lattice
Electromagnetic current:
𝑗 𝜇 ℓ𝑠 = 𝑗 𝜇 ℓ + 𝑗 𝜇 𝑠 = 𝑢 𝛾 𝜇 𝑢− 𝑑 𝛾 𝜇 𝑑 ( 𝑢 𝛾 𝜇 𝑢+ 𝑑 𝛾 𝜇 𝑑−2 𝑠 𝛾 𝜇 𝑠)
𝐺 𝑥 0 = 5 9 𝐺 con ℓ 𝑥 𝐺 con 𝑠 𝑥 𝐺 disc ℓ𝑠 ( 𝑥 0 )
[V. Lattice14]
Error on disconnected contribution dominates for 𝑥 0 ≳1.5 fm

29. Hadronic vacuum polarisation on the Lattice
Disconnected contribution for 𝑥 0 →∞:
𝐺 𝑥 0 = 𝐺 𝜌𝜌 𝑥 0 (1+𝑂 𝑒 − 𝑚 𝜋 𝑥 0 )
1 9 𝐺 𝑑isc ℓ𝑠 𝐺 𝜌𝜌 = 𝐺 𝑥 0 − 𝐺 𝜌𝜌 ( 𝑥 0 ) 𝐺 𝜌𝜌 ( 𝑥 0 ) − 𝐺 con 𝑠 𝑥 0 𝐺 𝑐on ℓ 𝑥 𝑥 0 →∞ − 1 9
Loss of signal at 𝑥 0 ≳1.5 fm provides upper bound on error
Disconnected contribution reduces Π ( 𝑄 2 ) by at most 2%

30. Summary Nucleon electromagnetic form factors and charge radii:
Large noise-to-signal ratio in baryonic correlation functions
Systematic effects may be hidden in the data
Good progress in reconciling experiment and lattice calculations
Strange form factors of the nucleon:
Lattice calculations mostly exploratory
Promising new techniques to evaluate disconnected diagrams
Requires huge statistics to address systematics
Hadronic vacuum polarisation contribution to the muon (g – 2):
Improve statistical accuracy
Better control over large- 𝑥 0 , low- 𝑄 2 regime required
Good prospects for quantifying disconnected contribution
Hadron form factors from O(a) improved Wilson quarks