Hadronic corrections to parity violation from the latice Jeremy Green Institut für Kernphysik, Johannes Gutenberg-Universität Mainz Physics beyond the standard model and precision nucleon structure measurements with parity-violating electron scatering ECT*, Trento, Italy August 1–5, 2016
Outline 1. Introduction 2. Latice QCD 3. Axial charge 4. Isovector electromagnetic form factors 5. Strange electromagnetic form factors 6. Light and strange axial form factors 7. Summary Jeremy Green (Mainz) Hadronic corrections to parity violation from the latice ECT*, August 1–5, 2016 2 / 27
Nucleon vector and axial form factors Describe the strength of the coupling of a proton to a current: � � 1 ( Q 2 ) + i σ µν ( p ′ − p ) ν � p ′ | V q γ µ F q F q u ( p ′ ) 2 ( Q 2 ) µ | p � = ¯ u ( p ) � 2 m p � A ( Q 2 ) + ( p ′ − p ) µ � p ′ | A q γ µ G q G q u ( p ′ ) P ( Q 2 ) µ | p � = ¯ γ 5 u ( p ) , 2 m p where V q q γ µ q and A q q γ µ γ 5 q . Electric and magnetic form factors: µ = ¯ µ = ¯ Q 2 G q E ( Q 2 ) = F q ( 2 m p ) 2 F q G q M ( Q 2 ) = F q 1 ( Q 2 ) + F q 1 ( Q 2 ) − 2 ( Q 2 ) , 2 ( Q 2 ) . Jeremy Green (Mainz) Hadronic corrections to parity violation from the latice ECT*, August 1–5, 2016 3 / 27
Electromagnetic form factors Elastic ep scatering has a leading contribution from single photon exchange. This allows the measurement of G γ ( p ) E , M = 2 3 G u E , M − 1 3 G d E , M − 1 3 G s E , M + . . . E = − 6 G γ ( p ) ′ and in particular the charge radius r 2 ( 0 ) , where experimental E results suffer from the proton radius problem . Assuming isospin, elastic en scatering probes the same, with u ↔ d . Finally, parity-violating elastic ep scatering is sensitive to the interference between photon and Z boson exchange. This can be used to determine 3 sin 2 θ W ) G u 3 sin 2 θ W )( G d G Z ( p ) E , M = ( 1 − 8 E , M − ( 1 − 4 E , M + G s E , M ) + . . . Jeremy Green (Mainz) Hadronic corrections to parity violation from the latice ECT*, August 1–5, 2016 4 / 27
Axial form factors Assuming isospin, the interaction between nucleons and a W boson contains the isovector axial current A u − d . µ ◮ Axial charge g u − d ≡ G u − d ( 0 ) = 1 . 2723 ( 23 ) measured from neutron beta A A decay. This has long served as a “benchmark” for latice QCD. ◮ Qasielastic neutrino scatering, e.g. ¯ ν e p → e + n , is sensitive to G u − d . A ◮ Muon capture, µ − p → ν µ n , is sensitive to G u − d . P The interaction between a proton and a Z boson contains the axial current A u − d − s . µ ◮ Relevant for elastic ν p and parity-violating elastic ep scatering. Jeremy Green (Mainz) Hadronic corrections to parity violation from the latice ECT*, August 1–5, 2016 5 / 27
Latice QCD ...is a regularization of Euclidean-space QCD such that the path integral can be done fully non-perturbatively. ◮ Euclidean spacetime becomes a periodic hypercubic latice, with spacing a and box size L 3 s × L t . ◮ Path integral over fermion degrees of freedom is done analytically, for each gauge configuration. Solving the Dirac equation with a fixed source yields a source-to-all quark propagator. ◮ Path integral over gauge degrees of freedom is done numerically using Monte Carlo methods to generate an ensemble of gauge configurations . The a → 0 and L s , L t → ∞ extrapolations need to be taken by using multiple ensembles. Jeremy Green (Mainz) Hadronic corrections to parity violation from the latice ECT*, August 1–5, 2016 6 / 27
Precision latice QCD Flavour Latice Averaging Group (FLAG) reviews: http://itpwiki.unibe.ch/flag continuum extrapolation G. Colangelo et al. , Eur. Phys. J. C 71 , 1695 (2011) chiral extrapolation S. Aoki et al. , Eur. Phys. J. C 74 , 2890 (2014) publication status renormalization S. Aoki et al. , 1607.00299 finite volume e.g., quark masses running Collaboration Ref. m ud m s RBC/UKQCD 14B ⊖ [10] P ⋆ ⋆ ⋆ ⋆ d 3.31(4)(4) 90.3(0.9)(1.0) RBC/UKQCD 12 ⊖ ⋆ ⋆ ⋆ [31] A ◦ d 3.37(9)(7)(1)(2) 92.3(1.9)(0.9)(0.4)(0.8) PACS-CS 12 ⋆ [143] A ⋆ ⋆ b 3.12(24)(8) 83.60(0.58)(2.23) � � ⋆ ⋆ Laiho 11 [44] C ◦ ◦ − 3.31(7)(20)(17) 94.2(1.4)(3.2)(4.7) BMW 10A, 10B + [7, 8] A ⋆ ⋆ ⋆ ⋆ c 3.469(47)(48) 95.5(1.1)(1.5) PACS-CS 10 [95] A ⋆ ⋆ b 2.78(27) 86.7(2.3) � � MILC 10A [13] C ⋆ ⋆ 3.19(4)(5)(16) – ◦ ◦ − HPQCD 10 ∗ [9] A ⋆ ⋆ − − 3.39(6) 92.2(1.3) ◦ ⋆ ⋆ RBC/UKQCD 10A [144] A ◦ ◦ a 3.59(13)(14)(8) 96.2(1.6)(0.2)(2.1) Blum 10 † [103] A ⋆ 3.44(12)(22) 97.6(2.9)(5.5) ◦ � ◦ − PACS-CS 09 [94] A ⋆ ⋆ b 2.97(28)(3) 92.75(58)(95) � � HPQCD 09A ⊕ [18] A ⋆ ⋆ 3.40(7) 92.4(1.5) ◦ − − MILC 09A [6] C ⋆ ⋆ − 3.25 (1)(7)(16)(0) 89.0(0.2)(1.6)(4.5)(0.1) ◦ ◦ ⋆ ⋆ MILC 09 [89] A ◦ ◦ − 3.2(0)(1)(2)(0) 88(0)(3)(4)(0) PACS-CS 08 [93] A ⋆ − 2.527(47) 72.72(78) � � � RBC/UKQCD 08 [145] A ◦ ⋆ ⋆ − 3 . 72(16)(33)(18) 107 . 3(4 . 4)(9 . 7)(4 . 9) � CP-PACS/ ⋆ ⋆ 3 . 55(19)( +56 90 . 1(4 . 3)( +16 . 7 [146] A � � − − 20 ) − 4 . 3 ) JLQCD 07 3 . 2(0)(2)(2)(0) ‡ 87(0)(4)(4)(0) ‡ HPQCD 05 [147] A ◦ ◦ ◦ ◦ − MILC 04, HPQCD/ [107, 148] A ◦ ◦ ◦ � − 2 . 8(0)(1)(3)(0) 76(0)(3)(7)(0) MILC/UKQCD 04 Jeremy Green (Mainz) Hadronic corrections to parity violation from the latice ECT*, August 1–5, 2016 7 / 27
Precision latice QCD Flavour Latice Averaging Group (FLAG) reviews: http://itpwiki.unibe.ch/flag G. Colangelo et al. , Eur. Phys. J. C 71 , 1695 (2011) S. Aoki et al. , Eur. Phys. J. C 74 , 2890 (2014) S. Aoki et al. , 1607.00299 + e.g., quark masses FLAG average for = + + + ETM 14 = = + FLAG average for RBC/UKQCD 14B RBC/UKQCD 12 PACS-CS 12 Laiho 11 + BMW 10A, 10B FLAG average for = + + PACS-CS 10 + MILC 10A HPQCD 14A + HPQCD 10 ETM 14 RBC/UKQCD 10A = = Blum 10 FLAG average for = + PACS-CS 09 RBC/UKQCD 14B HPQCD 09A RBC/UKQCD 12 MILC 09A PACS-CS 12 MILC 09 + BMW 10A, 10B PACS-CS 08 PACS-CS 10 RBC/UKQCD 08 = HPQCD 10 CP-PACS/JLQCD 07 RBC/UKQCD 10A HPQCD 05 MILC 04, HPQCD/MILC/UKQCD 04 Blum 10 PACS-CS 09 FLAG average for = HPQCD 09A MILC 09A Dürr 11 ETM 10B FLAG average for = JLQCD/TWQCD 08A ALPHA 12 RBC 07 = Dürr 11 ETM 07 = ETM 10B QCDSF/UKQCD 06 RBC 07 SPQcdR 05 ETM 07 QCDSF/UKQCD 04 QCDSF/UKQCD 06 JLQCD 02 CP-PACS 01 PDG pheno. pheno. Dominguez 09 PDG Chetyrkin 06 Dominguez 09 Jamin 06 Narison 06 Narison 06 Maltman 01 Vainshtein 78 2 3 4 5 6 MeV 70 80 90 100 110 120 MeV Jeremy Green (Mainz) Hadronic corrections to parity violation from the latice ECT*, August 1–5, 2016 7 / 27
Nucleon matrix elements using latice QCD To find matrix elements, compute using an interpolating operator χ : � e − i � p · � x � χ ( � χ ( � C 2pt ( t , � 0 , 0 ) � p ) = x , t ) ¯ � x t →∞ −→ e − E ( � p ) t |� p | ¯ χ | Ω �| 2 � p ′ · � p ′ − � p ′ ) = e − i � x e i ( � p ) · � y � χ ( � χ ( � C 3pt ( T , τ ; � p , � x , T ) O ( � 0 , 0 ) � y , τ ) ¯ x , � � y τ →∞ p ) τ � Ω | χ | p ′ �� p ′ |O| p �� p | ¯ T − τ →∞ p ′ )( T − τ ) e − E ( � −→ e − E ( � χ | Ω � Then form ratios to isolate � p ′ |O| p � . Jeremy Green (Mainz) Hadronic corrections to parity violation from the latice ECT*, August 1–5, 2016 8 / 27
������ ������ � � ��� ��� ��� ��� �� ������ ������ ������ ������ ������ ������ Systematic error: excited states With interpolating operator χ , compute, e.g., e − E n t � � � n | χ † | 0 � � � � 2 � � C 2pt ( t ) = � χ ( t ) χ † ( 0 ) � = n For a nucleon, the signal-to-noise asymptotically decays as e − ( m N − 3 2 m π ) t . Jeremy Green (Mainz) Hadronic corrections to parity violation from the latice ECT*, August 1–5, 2016 9 / 27
���� �� � � ���������� �� � �� � ��� ��� ��� �� ���� ���� ���� ���� ���� ���� ���� Systematic error: excited states With interpolating operator χ , compute, e.g., e − E n t � � � n | χ † | 0 � � � � 2 � � C 2pt ( t ) = � χ ( t ) χ † ( 0 ) � = n For a nucleon, the signal-to-noise asymptotically decays as e − ( m N − 3 2 m π ) t . Jeremy Green (Mainz) Hadronic corrections to parity violation from the latice ECT*, August 1–5, 2016 9 / 27
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