Ben Hörz (LBNL) Frontiers in Lattice QCD and related topics Yukawa Institute for Theoretical Physics, Kyoto University Apr 17, 2019 Progress towards nucleon-nucleon interactions with stochastic LapH
[Estabrooks, Martin 1975] [Protopopescu et al. 1973] nucleon-nucleon interactions nucleon-hyperon interactions multi-hadron state 1. What can we learn about the QCD spectrum from first principles? 2. Lattice QCD as a tool for nuclear physics [NuSTEC White Paper: Status and Challenges of Neutrino-Nucleus Scattering 1706.03621] 1/16 Hadron interactions from Lattice QCD ∆ ⟨ Nπ | J µ | N ⟩ N Σ , N Λ
[Estabrooks, Martin 1975] [Protopopescu et al. 1973] nucleon-nucleon interactions nucleon-hyperon interactions multi-hadron state 1. What can we learn about the QCD spectrum from first principles? 2. Lattice QCD as a tool for nuclear physics [NuSTEC White Paper: Status and Challenges of Neutrino-Nucleus Scattering 1706.03621] 1/16 Hadron interactions from Lattice QCD ∆ ⟨ Nπ | J µ | N ⟩ N Σ , N Λ
[Estabrooks, Martin 1975] [Protopopescu et al. 1973] nucleon-nucleon interactions nucleon-hyperon interactions multi-hadron state 1. What can we learn about the QCD spectrum from first principles? physics [NuSTEC White Paper: Status and Challenges of Neutrino-Nucleus Scattering 1706.03621] 1/16 Hadron interactions from Lattice QCD ∆ 2. Lattice QCD as a tool for nuclear ⟨ Nπ | J µ | N ⟩ N Σ , N Λ
[Estabrooks, Martin 1975] [Protopopescu et al. 1973] nucleon-nucleon interactions nucleon-hyperon interactions multi-hadron state 1. What can we learn about the QCD spectrum from first principles? physics [NuSTEC White Paper: Status and Challenges of Neutrino-Nucleus Scattering 1706.03621] 1/16 Hadron interactions from Lattice QCD ∆ 2. Lattice QCD as a tool for nuclear ⟨ Nπ | J µ | N ⟩ N Σ , N Λ
single particle in a periodic box two spinless particles in a periodic box [Lüscher ’86, ’91] 2/16 Scattering from Lattice QCD ⇝ ∆ E ∝ e − mL L ⇝ ∆ E ∝ a 0 /L 3 + O ( L − 4 ) L ⇒ ‘The Lüscher method’
see also [Hansen, Sharpe 1901.00483] [Briceño, Dudek, Young 1706.06223] 3/16 Review of formalism and results
4/16 partial wave [Morningstar, Bulava, Singha, Brett, Fallica, Hanlon, BH 1707.05817] on Github group theory worked out and publicly available – known functions (total angular mom.) 2-particle channel Two-particle Quantization Condition E L – FV spectrum M − 1 ( E L ) + F ( E L , L ) [ ] det = 0 M – 2-to-2 scatt. ampl. F
4/16 partial wave [Morningstar, Bulava, Singha, Brett, Fallica, Hanlon, BH 1707.05817] on Github group theory worked out and publicly available – known functions (total angular mom.) 2-particle channel Two-particle Quantization Condition E L – FV spectrum M − 1 ( E L ) + F ( E L , L ) [ ] det = 0 M – 2-to-2 scatt. ampl. F
5/16 [plot adapted from Bulava, Fahy, BH, Juge, Morningstar, Wong 1604.05593] same data difgerent way to plot [Meyer, Wittig 1807.09370] e.g. Lang et al. 1105.5636, Aoki et al. 1106.5365, …, Dudek et al. 1212.0830, … • benchmark system for the lattice A simple (yet relevant) resonance: ρ (770) 170 4 . 0 E ∗ /m π 130 3 . 5 δ 1 / ◦ 90 3 . 0 50 2 . 5 10 2 . 0 1 u (0) 1 (1) ) 1 (2) 1 (2) 2 (2) 1 (3) ) 2 . 0 2 . 5 3 . 0 3 . 5 4 . 0 1 3 ( ( E ∗ /m π A + E + A + B + B + A + E + ⇔ T + 10 ( q cm / m π ) 3 cot δ 1 • elastic ππ scattering neglecting ℓ ≥ 3 partial wave 5 spectrum ⇔ scattering amplitude 0 − 5 − 10 2 . 0 2 . 5 3 . 0 3 . 5 4 . 0 E ∗ /m π • recent interest due to its contribution to ( g − 2) µ HVP
5/16 [plot adapted from Bulava, Fahy, BH, Juge, Morningstar, Wong 1604.05593] same data difgerent way to plot [Meyer, Wittig 1807.09370] e.g. Lang et al. 1105.5636, Aoki et al. 1106.5365, …, Dudek et al. 1212.0830, … • benchmark system for the lattice A simple (yet relevant) resonance: ρ (770) 170 4 . 0 E ∗ /m π 130 3 . 5 δ 1 / ◦ 90 3 . 0 50 2 . 5 10 2 . 0 1 u (0) 1 (1) ) 1 (2) 1 (2) 2 (2) 1 (3) ) 2 . 0 2 . 5 3 . 0 3 . 5 4 . 0 1 3 ( ( E ∗ /m π A + E + A + B + B + A + E + ⇔ T + 10 ( q cm / m π ) 3 cot δ 1 • elastic ππ scattering neglecting ℓ ≥ 3 partial wave 5 spectrum ⇔ scattering amplitude 0 − 5 − 10 2 . 0 2 . 5 3 . 0 3 . 5 4 . 0 E ∗ /m π • recent interest due to its contribution to ( g − 2) µ HVP
5/16 [plot adapted from Bulava, Fahy, BH, Juge, Morningstar, Wong 1604.05593] same data difgerent way to plot [Meyer, Wittig 1807.09370] e.g. Lang et al. 1105.5636, Aoki et al. 1106.5365, …, Dudek et al. 1212.0830, … • benchmark system for the lattice A simple (yet relevant) resonance: ρ (770) 170 4 . 0 E ∗ /m π 130 3 . 5 δ 1 / ◦ 90 3 . 0 50 2 . 5 10 2 . 0 1 u (0) 1 (1) ) 1 (2) 1 (2) 2 (2) 1 (3) ) 2 . 0 2 . 5 3 . 0 3 . 5 4 . 0 1 3 ( ( E ∗ /m π A + E + A + B + B + A + E + ⇔ T + 10 ( q cm / m π ) 3 cot δ 1 • elastic ππ scattering neglecting ℓ ≥ 3 partial wave 5 spectrum ⇔ scattering amplitude 0 − 5 − 10 2 . 0 2 . 5 3 . 0 3 . 5 4 . 0 E ∗ /m π • recent interest due to its contribution to ( g − 2) µ HVP
5/16 [plot adapted from Bulava, Fahy, BH, Juge, Morningstar, Wong 1604.05593] same data difgerent way to plot [Meyer, Wittig 1807.09370] e.g. Lang et al. 1105.5636, Aoki et al. 1106.5365, …, Dudek et al. 1212.0830, … • benchmark system for the lattice A simple (yet relevant) resonance: ρ (770) 170 4 . 0 E ∗ /m π 130 3 . 5 δ 1 / ◦ 90 3 . 0 50 2 . 5 10 2 . 0 1 u (0) 1 (1) ) 1 (2) 1 (2) 2 (2) 1 (3) ) 2 . 0 2 . 5 3 . 0 3 . 5 4 . 0 1 3 ( ( E ∗ /m π A + E + A + B + B + A + E + ⇔ T + 10 ( q cm / m π ) 3 cot δ 1 • elastic ππ scattering neglecting ℓ ≥ 3 partial wave 5 spectrum ⇔ scattering amplitude 0 − 5 − 10 2 . 0 2 . 5 3 . 0 3 . 5 4 . 0 E ∗ /m π • recent interest due to its contribution to ( g − 2) µ HVP
5/16 [plot adapted from Bulava, Fahy, BH, Juge, Morningstar, Wong 1604.05593] same data difgerent way to plot [Meyer, Wittig 1807.09370] e.g. Lang et al. 1105.5636, Aoki et al. 1106.5365, …, Dudek et al. 1212.0830, … • benchmark system for the lattice A simple (yet relevant) resonance: ρ (770) 170 4 . 0 E ∗ /m π 130 3 . 5 δ 1 / ◦ 90 3 . 0 50 2 . 5 10 2 . 0 1 u (0) 1 (1) ) 1 (2) 1 (2) 2 (2) 1 (3) ) 2 . 0 2 . 5 3 . 0 3 . 5 4 . 0 1 3 ( ( E ∗ /m π A + E + A + B + B + A + E + ⇔ T + 10 ( q cm / m π ) 3 cot δ 1 • elastic ππ scattering neglecting ℓ ≥ 3 partial wave 5 spectrum ⇔ scattering amplitude 0 − 5 − 10 2 . 0 2 . 5 3 . 0 3 . 5 4 . 0 E ∗ /m π • recent interest due to its contribution to ( g − 2) µ HVP
5/16 [plot adapted from Bulava, Fahy, BH, Juge, Morningstar, Wong 1604.05593] same data difgerent way to plot [Meyer, Wittig 1807.09370] e.g. Lang et al. 1105.5636, Aoki et al. 1106.5365, …, Dudek et al. 1212.0830, … • benchmark system for the lattice A simple (yet relevant) resonance: ρ (770) 170 4 . 0 mπ =233 MeV E ∗ /m π 130 3 . 5 δ 1 / ◦ 90 3 . 0 50 2 . 5 10 2 . 0 1 u (0) 1 (1) ) 1 (2) 1 (2) 2 (2) 1 (3) ) 2 . 0 2 . 5 3 . 0 3 . 5 4 . 0 1 3 ( ( E ∗ /m π A + E + A + B + B + A + E + ⇔ T + 10 ( q cm / m π ) 3 cot δ 1 • elastic ππ scattering neglecting ℓ ≥ 3 partial wave 5 spectrum ⇔ scattering amplitude 0 − 5 − 10 2 . 0 2 . 5 3 . 0 3 . 5 4 . 0 E ∗ /m π • recent interest due to its contribution to ( g − 2) µ HVP
5/16 [plot adapted from Bulava, Fahy, BH, Juge, Morningstar, Wong 1604.05593] same data difgerent way to plot [Meyer, Wittig 1807.09370] e.g. Lang et al. 1105.5636, Aoki et al. 1106.5365, …, Dudek et al. 1212.0830, … • benchmark system for the lattice A simple (yet relevant) resonance: ρ (770) 170 4 . 0 mπ =233 MeV E ∗ /m π 130 3 . 5 δ 1 / ◦ 90 3 . 0 50 2 . 5 10 2 . 0 1 u (0) 1 (1) ) 1 (2) 1 (2) 2 (2) 1 (3) ) 2 . 0 2 . 5 3 . 0 3 . 5 4 . 0 1 3 ( ( E ∗ /m π A + E + A + B + B + A + E + ⇔ T + 10 ( q cm / m π ) 3 cot δ 1 • elastic ππ scattering neglecting ℓ ≥ 3 partial wave 5 spectrum ⇔ scattering amplitude 0 − 5 − 10 2 . 0 2 . 5 3 . 0 3 . 5 4 . 0 E ∗ /m π • recent interest due to its contribution to ( g − 2) µ HVP
6/16 Lellouch, Lüscher hep-lat/0003023 scattering amplitude QCD requires finite volume • HVP governed by infinite volume Feng et al. 1412.6319 • two-pion state dominates at low energies Meyer 1105.1892 Matrix elements: Timelike pion form factor • muon anomalous magnetic moment ( g − 2) µ γ R had = σ ( e + e − → hadrons) / 4 πα em ( s ) 2 3 s ) 3 ( 1 − 4 m 2 2 µ µ | F π ( s ) | 2 R had ( s ) = 1 π 4 s γ ∗ → ππ | F π ( E ∗ ) | 2 = g Λ ( γ ) q ∂ ( δ 1 + F ) � 2 3 πE ∗ 2 � ⟨ 0 | V ( d , Λ) | d Λ E ∗ ⟩ � � 2 q 5 L 3 ∂q
6/16 Lellouch, Lüscher hep-lat/0003023 scattering amplitude QCD requires finite volume • HVP governed by infinite volume Feng et al. 1412.6319 • two-pion state dominates at low energies Meyer 1105.1892 Matrix elements: Timelike pion form factor • muon anomalous magnetic moment ( g − 2) µ γ R had = σ ( e + e − → hadrons) / 4 πα em ( s ) 2 3 s ) 3 ( 1 − 4 m 2 2 µ µ | F π ( s ) | 2 R had ( s ) = 1 π 4 s γ ∗ → ππ | F π ( E ∗ ) | 2 = g Λ ( γ ) q ∂ ( δ 1 + F ) � 2 3 πE ∗ 2 � ⟨ 0 | V ( d , Λ) | d Λ E ∗ ⟩ � � 2 q 5 L 3 ∂q
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