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

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] 19.11.2018

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) ⟶ 19.11.2018

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] 19.11.2018

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 19.11.2018

I. Nucleon Form Factors and Charge Radii 19.11.2018

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. Constantinou @ 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 19.11.2018

Statistical fluctuations in baryon correlators Noise-to-signal ratio increases exponentially: Nucleon at rest: 𝑅 NS 𝑥 0 ∼ e 𝑚 N − 3 2 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 19.11.2018

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 19.11.2018

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 ′ 19.11.2018

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:1311.5804, and in prep.] 19.11.2018

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., 1404.3723] Summation method: 𝑆 𝑉 ( 𝑞 , 𝑡 𝑠 )= 𝐾 V + 𝑡 𝑠 𝐺 E,M ( 𝑄 2 ) [Jäger, Rae et al., arXiv:1311.5804, and in prep.] 19.11.2018

Methods for determining form factors Chiral behaviour of 𝐺 𝐸 at 𝑄 2 =0.1 GeV: [Jäger, Rae et al., arXiv:1311.5804, and in prep.] 19.11.2018

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:1310.7043] 19.11.2018

II. Quark-disconnected Diagrams 19.11.2018

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 19.11.2018

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 19.11.2018

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: 19.11.2018

Strange form factors Doi et al., Phys Rev D80 (2009) 094503 Two-flavour QCD; 𝑂 𝑎 improved Wilson quarks; 16 3 ⋅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 19.11.2018

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 =−0.015 23 , 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 19.11.2018

Strange form factors Babich et al., Phys Rev D85 (2012) 054510 Two-flavour QCD; 𝑂 𝑎 improved Wilson fermions; 24 3 ⋅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 19.11.2018

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) 19.11.2018

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 19.11.2018

III. Further Applications 19.11.2018

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 19.11.2018

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) 094503] Fit to ChPT @ NLO: 𝑟 2 s =0.635± 0.016 stat f m 2 Inclusion of disconnected part crucial for agreement with 𝜋𝜋-scattering 19.11.2018

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 19.11.2018

Hadronic vacuum polarisation on the Lattice Electromagnetic current: 𝑗 𝜇 ℓ𝑠 = 𝑗 𝜇 ℓ + 𝑗 𝜇 𝑠 = 1 2 𝑢 𝛾 𝜇 𝑢− 𝑑 𝛾 𝜇 𝑑 + 1 6 ( 𝑢 𝛾 𝜇 𝑢+ 𝑑 𝛾 𝜇 𝑑−2 𝑠 𝛾 𝜇 𝑠) 𝐺 𝑥 0 = 5 9 𝐺 con ℓ 𝑥 0 + 1 9 𝐺 con 𝑠 𝑥 0 + 1 9 𝐺 disc ℓ𝑠 ( 𝑥 0 ) [V. Gülpers @ Lattice14] Error on disconnected contribution dominates for 𝑥 0 ≳1.5 fm 19.11.2018

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 ) − 1 9 1+2 𝐺 con 𝑠 𝑥 0 𝐺 𝑐on ℓ 𝑥 0 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% 19.11.2018

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 19.11.2018