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Numerical Evaluation of QED Contribution to Lepton G−2

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Numerical Evaluation of QED Contribution to Lepton G−2 Numerical evaluation of QED contribution to lepton g−2 T. Aoyama KMI, Nagoya University The 33rd International Symposium on Lattice Field Theory (LATTICE 2015) July 14–18, 2015, Kobe Anomalous magnetic moment of lepton ◮ Electrons and Muons have magnetic moment along their spins, given by e~ ~µ = g ~s 2m It is known that g-factor deviates from Dirac’s value (g = 2), and it is called Anomalous magnetic moment aℓ (g 2)/2 ≡ − It is much precisely measured for electron and muon. ◮ Electron g 2 is explained almost entirely by QED interaction between electron and− photons. It has provided the most stringent test of QED. ◮ Muon g 2 is more sensitive to high energy physics, and thus a window to new physics− beyond the standard model. 1/55 10th-order project ◮ Numerical evaluation of the entire 10th order QED contribution to lepton g 2 has been conducted by the collaboration with: − Toichiro Kinoshita (Cornell, and UMass Amherst) Makiko Nio (RIKEN) Masashi Hayakawa (Nagoya University) Noriaki Watanabe (Nagoya University) Katsuyuki Asano (Nagoya University) 2/55 Electron g−2 figure from the slide by S. Fogwell Hoogerheide at Lepton moments 2014 Anomalous magnetic moment of electron ◮ Latest measurement by Harvard group by the resonance of cyclotron and spin levels for a single electron in a cylindrical Penning trap: ae(HV06) = 0.001 159 652 180 85 (76) [0.66ppb] Odom, Hanneke, D’Urso, Gabrielse, PRL97, 030801 (2006) ae(HV08) = 0.001 159 652 180 73 (28) [0.24ppb] Hanneke, Fogwell, Gabrielse, PRL100, 120801 (2008) Hanneke, Fogwell Hoogerheide, Gabrielse, PRA83, 052122 (2011) trap cavity electron top endcap electrode quartz spacer compensation electrode nickel rings ring electrode 0.5 cm compensation bottom endcap electrode electrode field emission microwave inlet point FIG. 2 (color). Cylindrical Penning trap cavity used to confine a single electron and inhibit spontaneous emission. ◮ This result is 15-fold improvement over the previous measurement by the University of Washington group: ae− (UW87) = 0.001 159 652 188 4 (43) [3.7ppb] ae+ (UW87) = 0.001 159 652 187 9 (43) [3.7ppb] Van Dyck, Schwinberg, Dehmelt, PRL59, 26 (1987) 3/55 Standard Model prediction of ae ◮ Contributions to electron g 2 within the context of the standard model consist of: − ae = ae(QED) + ae(Hadronic) + ae(Weak) ◮ QED contribution can be written as ae(QED) = A1 + A2(me/mµ) + A2(me/mτ ) + A3(me/mµ, me/mτ ) mass-dependent contribution ◮ Current status of the standard model prediction is: ae(QED mass-independent) 0.001 159 652 177 160 (763) −12 ae(QED mass-dependent) 2.748 (2) 10 × −12 ae(Hadronic) 1.705 (11) 10 × −12 ae(Weak) 0.0297 (5) 10 × ae(theory) 0.001 159 652 181 643 (763) ∗ Uncertainty comes from that of fine-structure constant α. 4/55 QED contribution ◮ QED contributions are evaluated by perturbation theory: 2 3 4 (2) α (4) α (6) α (8) α aℓ(QED) = A + A + A + A + π π π π ··· ◮ Up to which order of the QED perturbation theory do we need, to meet the precision of the measurements? α 4 29.1 10−12, π ≃ × α 5 0.067 10−12, π ≃ × the experimental uncertainty −12 δae(exp) = 0.28 10 . × ◮ Therefore, we need to know A(8) up to O(10−3), and A(10) up to O(10−1). 5/55 QED contribution: 2nd order term ◮ 2nd order term comes from 1 Feynman diagram: 1 A(2) = 1 2 Schwinger, PR73, 416 (1948) ◮ With this result, the electron g 2 up to 2nd order becomes: − (2) ae = 0.001 161 ... which well explained the observed value in the study of Zeeman splitting of gallium atom by Kusch and Foley in 1947, ae(KF47) = 0.001 19 (5) Kusch and Foley, PR72, 1256 (1947); PR74, 250 (1948) 6/55 QED contribution: 4th order term ◮ 4th order term comes from 7 Feynman diagrams: Their contributions are known analytically: 197 1 3 A(4) = + 3ln2 ζ(2) + ζ(3) 1 144 2 − 4 = 0.328 478 965 579 ... − Petermann, Helv.Phys.Acta 30, 407 (1957) Sommerfield, PR107, 328 (1957) ◮ 1 diagram with muon or tau-lepton loop also contributes to mass-dependent A2 terms. Their numerical values are: (4) −7 A (me/mµ) = 5.197 386 67 (26) 10 2 × (4) −9 A (me/mτ ) = 1.837 98 (34) 10 2 × Elend, PL20, 682 (1966) Samuel and Li, PRD44, 3935 (1991); 46, 4782(E) (1993); Li, Mendel, and Samuel, PRD47, 1723 (1993) Passera, J.Phys.G31, R75 (2005) where the values of the mass ratios used are: −3 −4 me/mµ = 4.836 331 66 (12) × 10 , me/mτ = 2.875 92 (26) × 10 . Mohr, Taylor Newell, Rev.Mod.Phys.84, 1527 (2012) [CODATA2010] 7/55 QED contribution: 6th order term ◮ 6th order term receives contributions from 72 Feynman diagrams, represented by these five types: Their contributions are analytically known, after almost 30 years of works that completed in late 1990’s. The numerical values are: (6) A1 = 1.181 241 456 ... (6) A (me/mµ) = 0.000 007 373 941 62 (27) 2 − (6) A (me/mτ ) = 0.000 000 065 830 (11) 2 − (6) A3 (me/mµ, me/mτ ) = 0.000 000 000 000 190 9 (1) Magnaco and Remiddi, Nuovo Cim.A60, 519 (1969) Barbieri, Remiddi, PLB49, 468 (1974); Barbieri, Caffo, and Remiddi, PLB57, 460 (1975) Levine, Remiddi, and Roskies, PRD20, 2068 (1979); Laporta and Remiddi, PLB265, 182 (1991); 390, 390 (1995) Laporta, PRD47, 4793 (1993); PLB343, 421 (1995) Laporta and Remiddi, PLB379, 283 (1996) Laporta, Nuovo Cim.A106, 675 (1993); Laporta and Remiddi, PLB301, 440 (1993) 8/55 QED contribution: 6th order term ◮ Numerical evaluation of these 6th order contributions were started in late 1960’s. The early results led to: (6) A1 = 1.195 (26) Kinoshita and Cvitanovic,´ PRD10, 4007 (1974) cf. Levine and Wright, PRD8, 3171 (1973) The best numerical value of the diagram M6H combined with the analytical values of the other diagrams was: (6) A1 = 1.181 259 (40) Kinoshita, PRL75, 4728 (1995) which agrees with the analytical result to 5 digits. ◮ Formulation of the numerical evaluation developed for the 6th order calculation has been extended for application to the 8th order, and the present 10th order calculations. 9/55 QED contribution: 8th order term ◮ There are 891 Feynman diagrams contributing to 8th order term. They are classified into 13 gauge-invariant groups. I(a) I(b) I(c) I(d) II(a) II(b) II(c) III IV(a) IV(b) IV(c) IV(d) V ◮ They are mostly evaluated by numerical means. The latest result of the (8) mass-independent term A1 is A(8) = 1.912 98 (84) 1 − Caffo, Turrini, Remiddi, PRD30, 483 (1984) Remiddi, Sorella, Lett.Nuovo Cim.44, 231 (1985) Kinoshita and Lindquist, PRD27, 867 (1983); PRD27, 877 (1987); PRD27, 886 (1983); PRD39, 2407 (1989); PRD42, 636 (1990) Kinoshita and Nio, PRL90, 021803 (2003) Kinoshita and Nio, PRD73, 013003 (2006) TA, Hayakawa, Kinoshita, Nio, PRL99, 110406 (2007); PRD77, 053012 (2008) TA, Hayakawa, Kinoshita, Nio, PRL109, 111807 (2012) TA, Hayakawa, Kinoshita, Nio, PRD91, 033006 (2015) 10/55 QED contribution: 8th order term ◮ (8) (8) Mass-dependent terms A2 (me/mµ), A2 (me/mτ ), and (8) A3 (me/mµ, me/mτ ) are also evaluated numerically: (8) A2 (me/mµ) = 0.000 922 (66) (8) A2 (me/mτ ) = 0.000 008 24 (12) 0.000 007 38 (12)∗ → (8) A3 (me/mµ, me/mτ ) = 0.000 000 746 5 (18) TA, Hayakawa, Kinoshita, Nio, PRL109, 111807 (2012) (∗ after correcting the typo found in the literature) ◮ Recently these terms were evaluated by analytic method in heavy lepton-mass expansion. (8) A2 (me/mµ) = 0.000 916 197 070 3 (373) (8) A2 (me/mτ ) = 0.000 007 429 24 (118) (8) A3 (me/mµ, me/mτ ) = 0.000 000 746 87 (28) Kurz, Liu, Marquard, Steinhauser, NPB879, 1 (2014) The agreement of these two values confirmed our numerical results. 11/55 QED contribution: 10th order term ◮ 12 672 Feynman diagrams contribute to 10th order term. They are classified into 32 gauge invariant sets within 6 supersets. 12/55 QED contribution: 10th order term ◮ Numerical evaluation of the complete 10th order contribution was reported in 2012 and an updated result was published in 2015. (10) A1 = 7.795 (336) Kinoshita and Nio, PRD73, 053007 (2006) TA, Hayakawa, Kinoshita, Nio, Watanabe, PRD78, 053005 (2008) TA, Asano, Hayakawa, Kinoshita, Nio, Watanabe, PRD81, 053009 (2010) TA, Hayakawa, Kinoshita, Nio, PRD78, 113006 (2008); 82, 113004 (2010); 83, 053002 (2011) 83, 053003 (2011); 84, 053003 (2011); 85, 033007 (2012); 85, 093013 (2012) TA, Hayakawa, Kinoshita, Nio, PRL109, 111807 (2012); PRD91, 033006 (2015) ◮ (10) Contribution to A1 mainly comes from Set V that consists of 6354 vertex diagrams without closed lepton loops. ◮ Mass-dependent term is also evaluated: (10) A (me/mµ) = 0.003 82 (39) 2 − tau-lepton contribution is negligibly small for the current experimental precision. 13/55 QED contribution: 10th order term ◮ Recently, Set I that consists of 208 diagrams are evaluated (semi-) analytically for mass-dependent and mass-independent contributions. Baikov, Maier, Marquard, NPB877, 647 (2013) I(a) I(b) I(c) I(d) I(e) I(f) I(g) I(h) I(i) I(j) ◮ (10) Their results of A1 [Set I] are consistent with our numerical evaluations: Baikov et al. AHKN, PRL2012 I(a) 0.000 471 0.000 470 94 (6) I(b) 0.007 010 0.007 010 8 (7) I(c) 0.023 467 0.023 468 (2) I(d) + I(e) 0.014 094 0.014 098 (5)(4) I(e) 0.010 291 0.010 296 (4) I(f) + I(g) + I(h) 0 037 85+5 0 037 833 20 6 13 .
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