Tobias G ocke yo 2. A Together with C. S. Fischer (JLU) and R. - PowerPoint PPT Presentation

Bad Honnef 2012 Hadronic con- tribution to the 1. Intro- 4. Summary muon g − 2 from duction and Outlook a Dyson-Schwinger perspective Tobias G¨ ocke yo 2. A Together with C. S. Fischer (JLU) and R. Williams (Complutense, Madrid) Functional 3. Results yo Approach [Fischer, TG, Williams, arXiv:1009.5297] yo [TG, Fischer, Williams, arXiv:1012.3886] yo [TG, Fischer, Williams, arXiv:1107.2588] JUSTUS-LIEBIG- UNIVERSITÄT GIESSEN Tobias G¨ ocke (JLU Gießen) Hadr. contr. to a µ from a DSE perspective 15.02.2012 1 / 17

Hadronic con- tribution to the 1. Intro- 4. Summary muon g − 2 from duction and Outlook a Dyson-Schwinger perspective 2. A Functional 3. Results Approach

Hadronic con- tribution to the 1. Intro- 4. Summary muon g − 2 from duction and Outlook a Dyson-Schwinger perspective 2. A Functional 3. Results Approach

1. Introduction a µ is. . . Precisely determined by experiment and accurately predicted by Standard Model Precision test for the Standard Model More sensitive to contributions from high scales than a e since : m µ ≫ m e Sensitive to QCD and potential ’new physics’ contributions Deviation between experiment and theory? So one has to . . . get the SM predictions under control use non-perturbative methods for QCD-contributions. Tobias G¨ ocke (JLU Gießen) Hadr. contr. to a µ from a DSE perspective 15.02.2012 4 / 17

1. Introduction Magnetic moment � µ , g -factor µ = g e � � S 2 m Dirac equation: g = 2 Schwinger: anomalous part a µ = g − 2 ≈ α/ 2 π ≈ 0 . 00116 2 Relativistic QFT � F 1 ( q 2 ) γ α + iF 2 ( q 2 ) σ αβ q β � = ( − ie )¯ u ( p ′ ) u ( p ) 2 m µ a µ := F 2 (0) = g µ − 2 2 Tobias G¨ ocke (JLU Gießen) Hadr. contr. to a µ from a DSE perspective 15.02.2012 5 / 17

1. Introduction Precision tests of the standard model a µ [10 − 11 ] contribution Experiment 116 592 089(63) SM 116 591 828(49) hadronic LO 6 949(43) hadronic LbL 105(26) exp. - th. 261(80) [B. L. Roberts, arXiv:1001.2898 (2010)] [Hagiwara, Liao, Martin, Nomura, Teubner, arXiv:1105.3149 (2011)] New Physics? 3.3 σ effect Lattice? First results but needs time Tobias G¨ ocke (JLU Gießen) Hadr. contr. to a µ from a DSE perspective 15.02.2012 6 / 17

1. Introduction Precision tests of the standard model a µ [10 − 11 ] contribution Experiment 116 592 089(63) SM 116 591 828(49) hadronic LO 6 949(43) hadronic LbL 105(26) exp. - th. 261(80) [B. L. Roberts, arXiv:1001.2898 (2010)] [Hagiwara, Liao, Martin, Nomura, Teubner, arXiv:1105.3149 (2011)] New Physics? 3.3 σ effect Lattice? First results but needs time Tobias G¨ ocke (JLU Gießen) Hadr. contr. to a µ from a DSE perspective 15.02.2012 6 / 17

1. Introduction Two Hadronic Contributions hadronic vacuum polarization hadr. light by light scattering one-scale problem two-scale problem known from dispersion only accessible through relation theory leading contribution systematic uncertainty? How to deal with these objects? − → calculate both from the same approach use the ’little brother’ as test case Tobias G¨ ocke (JLU Gießen) Hadr. contr. to a µ from a DSE perspective 15.02.2012 7 / 17

Hadronic con- tribution to the 1. Intro- 4. Summary muon g − 2 from duction and Outlook a Dyson-Schwinger perspective 2. A Functional 3. Results Approach

2. A Functional Approach - Overview Diagrammatic representation = = + q Model of quark-gluon interaction: [Maris and Roberts, arXiv:nucl-th/9708029] [Maris and Tandy, arXiv:nucl-th/9905056] [Maris and Tandy, arXiv:nucl-th/9910033] one description for high and low energy → important for two-scale problem building blocks Tobias G¨ ocke (JLU Gießen) Hadr. contr. to a µ from a DSE perspective 15.02.2012 9 / 17

Hadronic con- tribution to the 1. Intro- 4. Summary muon g − 2 from duction and Outlook a Dyson-Schwinger perspective 2. A Functional 3. Results Approach

3. Results - Hadronic Vacuum Polarization δ µν q 2 − q µ q ν � � Π( q 2 ) = Π µν ( q ) = Adler function: D ( q ) = − d Π / d ln q 2 two parameter sets, u and d quark masses fixed to the I:pion mass II: rho mass use u, d, s, c, b quarks (isospin-limit: m u = m d ) all parameters fixed by meson phenomenology ◮ model interaction → � ¯ ψψ � and f π ◮ m u / d → m π or m ρ ◮ m s → m K or m φ c and b ¯ ◮ m c / b → c ¯ b vector states Tobias G¨ ocke (JLU Gießen) Hadr. contr. to a µ from a DSE perspective 15.02.2012 11 / 17

3. Results - Hadronic Vacuum Polarization δ µν q 2 − q µ q ν � � Π( q 2 ) = Π µν ( q ) = Adler function: D ( q ) = − d Π / d ln q 2 two parameter sets, u and d quark masses fixed to the I:pion mass II: rho mass use u, d, s, c, b quarks (isospin-limit: m u = m d ) all parameters fixed by meson phenomenology ◮ model interaction → � ¯ ψψ � and f π ◮ m u / d → m π or m ρ ◮ m s → m K or m φ c and b ¯ ◮ m c / b → c ¯ b vector states Tobias G¨ ocke (JLU Gießen) Hadr. contr. to a µ from a DSE perspective 15.02.2012 11 / 17

3. Results - Hadronic Vacuum Polarization δ µν q 2 − q µ q ν � � Π( q 2 ) = Π µν ( q ) = Adler function: D ( q ) = − d Π / d ln q 2 two parameter sets, u and d quark masses fixed to the I:pion mass II: rho mass use u, d, s, c, b quarks (isospin-limit: m u = m d ) all parameters fixed by meson phenomenology ◮ model interaction → � ¯ ψψ � and f π ◮ m u / d → m π or m ρ ◮ m s → m K or m φ c and b ¯ ◮ m c / b → c ¯ b vector states Tobias G¨ ocke (JLU Gießen) Hadr. contr. to a µ from a DSE perspective 15.02.2012 11 / 17

3. Results - Hadronic Vacuum Polarization δ µν q 2 − q µ q ν � � Π( q 2 ) = Π µν ( q ) = Adler function: D ( q ) = − d Π / d ln q 2 two parameter sets, u and d quark masses fixed to the I:pion mass II: rho mass use u, d, s, c, b quarks (isospin-limit: m u = m d ) all parameters fixed by meson phenomenology ◮ model interaction → � ¯ ψψ � and f π ◮ m u / d → m π or m ρ ◮ m s → m K or m φ c and b ¯ ◮ m c / b → c ¯ b vector states Tobias G¨ ocke (JLU Gießen) Hadr. contr. to a µ from a DSE perspective 15.02.2012 11 / 17

3. Results - Hadronic Vacuum Polarisation Adler function D(Q) 0.03 0.02 0.01 dispersion relation DSE 0 0 2 4 6 8 10 Q [GeV] = (6760 − 7440) × 10 − 11 | a HVP , disp . rel . a HVP , DSE = 6903 . 0(52 . 6) × 10 − 11 µ µ [TG, Fischer, Williams, arXiv:1107.2588 (2011)] [Jegerlehner and Nyffeler ,arXiv:0902.3360 (2009)] Tobias G¨ ocke (JLU Gießen) Hadr. contr. to a µ from a DSE perspective 15.02.2012 12 / 17

3. Results - LbL - π , η , η ′ Pole pseudoscalar (PS) exchance π only a π − pole = 58(7) × 10 − 11 µ π , η and η ′ a PS − pole = 80 . 7(12) × 10 − 11 µ [Fischer, TG, Williams, arXiv:1009.5297] [TG, Fischer, Williams, arXiv:1012.3886] good agreement with existing approaches Tobias G¨ ocke (JLU Gießen) Hadr. contr. to a µ from a DSE perspective 15.02.2012 13 / 17

3. Results - LbL - Quark Loop quark loop bare vertex γ µ a LBL , bare = 61(2) × 10 − 11 µ γ µ with 1st Ball-Chiu dressing a LBL , 1 BC = 107(2) × 10 − 11 µ Only ’full’ self-consistent vertex from BSE will be conclusive → numerically quite challenging [Fischer, TG, Williams, arXiv:1009.5297] [TG, Fischer, Williams, arXiv:1012.3886] enhancement compared to existing approaches Tobias G¨ ocke (JLU Gießen) Hadr. contr. to a µ from a DSE perspective 15.02.2012 14 / 17

Hadronic con- tribution to the 1. Intro- 4. Summary muon g − 2 from duction and Outlook a Dyson-Schwinger perspective 2. A Functional 3. Results Approach

5. Summary and Outlook summary DSE calculation of g − 2 HVP ◮ a HVP , DSE = (6760 − 7440) × 10 − 11 µ ◮ a HVP , disp . rel . = 6949(43) × 10 − 11 µ ◮ Adler function can be described reasonably LbL ◮ a PS − pole = 81(12) × 10 − 11 µ ◮ enhancement of quark-loop overlooked? ◮ but no final answer yet [Hagiwara, Liao, Martin, Nomura, Teubner, arXiv:1105.3149 (2011)] Tobias G¨ ocke (JLU Gießen) Hadr. contr. to a µ from a DSE perspective 15.02.2012 16 / 17

5. Summary and Outlook outlook complete quark-loop calculation overcome resonance expansion ( π 0 , η, η ′ ) pole dominance Thank You for the attention! supported by DFG under grant No. Fi 970/8-1 Helmholtz Young Investigator Grant No. VH-NG-332 Helmholtz International Center for FAIR within the LOEWE program of the State of Hesse Science Fund FWF under Project No. P20592-N16 Ministerio de Educaci´ on (Spain): Programa Nacional de Movilidad de Recursos Humanos del, Plan Nacional de I-D+i 2008-2011 Tobias G¨ ocke (JLU Gießen) Hadr. contr. to a µ from a DSE perspective 15.02.2012 17 / 17

5. Summary and Outlook outlook complete quark-loop calculation overcome resonance expansion ( π 0 , η, η ′ ) pole dominance Thank You for the attention! supported by DFG under grant No. Fi 970/8-1 Helmholtz Young Investigator Grant No. VH-NG-332 Helmholtz International Center for FAIR within the LOEWE program of the State of Hesse Science Fund FWF under Project No. P20592-N16 Ministerio de Educaci´ on (Spain): Programa Nacional de Movilidad de Recursos Humanos del, Plan Nacional de I-D+i 2008-2011 Tobias G¨ ocke (JLU Gießen) Hadr. contr. to a µ from a DSE perspective 15.02.2012 17 / 17