Hadronic contributions to the muon g-2

Transcription

1 Hadronic contributions to the muon g-2 RICHARD WILLIAMS (HIRSCHEGG 2014) 1

2 Overview Introduction Hadronic Vacuum Polarisation Hadronic Light-by-Light Scattering Conclusions 2

3 Overview Introduction Hadronic Vacuum Polarisation Hadronic Light-by-Light Scattering Conclusions 3

4 Anomalous magnetic moment of the muon μ = g l eħ 2m l c S g l : Gyromagnetic ratio for spin 1 2 particle Dirac: g = 2 QFT: g > 2 (screening: Effective n-body interaction) a l = 1 (g 2) 2 Quantum corrections due to heavy particles δa l a l m l 2 m e 2 m 2 l M 2 : Muon more sensitive by ~200 2

5 Experimental Measurement: muon Larmor Procession B field exerts torque on the magnetic moment [Jegerlehner & Nyffeler, 2009] a μ = [MaNEP] [G. W. Benett et al (E821 at BNL), PRD 73, (2006)] 5

6 Muon Theory Budget QCD dominates theory error New physics = physics beyond QED + Weak QED/Weak perturbative QCD non-perturbative Contribution a μ [ ] % QED ( 0.2) Weak ( 1.8) QCD LOHVP (58.2) QCD HOHVP ( 1.0) QCD HLBL 105 (26 ) SM (64) 100 Experiment (63) - Difference 262 (89) - 6

7 QCD Corrections Hadronic Vacuum Polarisation Related to data (e+ e- annihilation) Early lattice QCD calculations Model calculations Hadronic light-by-light scattering No independent experimental input Very difficult lattice QCD calculation Model calculations: systematics? 7

8 Non-perturbative tools Effective Field Theories 1 N c counting: ENJL, LMD, CQM Chiral counting: ChPT Functional Methods Lattice QCD FRG/DSE/BSE Can apply to both LO and NLO Hadronic contributions 8

9 Dyson-Schwinger Equations Quark propagator DSE Quark-Photon Vertex DSE Integral Equations for Green s Functions of QFT 9

10 Overview Introduction Hadronic Vacuum Polarisation Hadronic Light-by-Light Scattering Conclusions 10

11 Hadronic Vacuum Polarisation Π μν : satisfies dispersion relation a LO,HVP μ = 1 0 4π 3 ds σ had s K(s) m2 π [Davier et al, EPJ C71 (2011) 1515] Direct scan SND/BES II/CMD Radiative Return KLOE/BELLE/BaBar e + e 2π Provides ~70% 11

12 Hadronic Vacuum Polarisation Data Extraction LO: 6951 ± HO: 98 ± LO: ± HO: 97.9 ± [Hagiwari et al, JPG 38 (2011) ] [Davier et al, EPJ C71 (2011) 1515] HLS eff. Interaction / global fit LO: φ ± [Benayoun et al, EPJ C73 (2013) 2453] Limited by availability / quality of data Systematics under control? Reasonable errors? 12

13 Hadronic Vacuum Polarisation: Theory Photon polarisation tensor / two-current correlator < > Π μν q = d 4 x e i q x j μ x j ν 0 1PI,hadr Quark current: j μ = 2 3 uγ μu 1 3 dγ μd 1 3 sγ μs + WI implies transversality Π μν q = δ μν q μq ν q 2 q 2 Π(q 2 ) 13

14 Hadronic Vacuum Polarisation: Lattice N f = 2 experiment a μ = Linear fit a μ = Vector meson fit a μ = [Jansen et al, 2011 (ETMC)] N f = a μ = (32) [Boyle et al, 2011 (RBC-UKQCD)] 14

15 Hadronic Vacuum Polarisation: DSE Exact equation Corresponds to Caveat Requires 1PI quark propagator and quark-photon vertex In turn require: Quark-antiquark interaction Quark-gluon vertex Gluon propagator... 15

16 Hadronic Vacuum Polarisation: DSE [dispersion data, Jegerlehner 2008] Adler function dπ q2 D q = q2 dq 2 [Goecke, Fischer and RW, PLB 704 (2011) ] a μ HVP = α π 0 1 dx(1 x) e 2 Π x 2 1 x m μ 2 16

17 Hadronic Vacuum Polarisation: DSE DSE (5 flavours) a μ HVP = m π mass fit a μ HVP = m ρ mass fit cf. experiment a μ HVP =

18 Overview Introduction Hadronic Vacuum Polarisation Hadronic Light-by-Light Scattering Conclusions 18

19 Hadronic light-by-light Scattering 19

20 Hadronic light-by-light Scattering

21 Hadronic light-by-light Scattering Evaluated at spacelike Euclidean momentum Time-like poles p 2 > 0 All bound-states/resonances are off-shell 21

22 Hadronic light-by-light Scattering [Melnikov et al, PRD 70 (2004)] [Engel et al, PRD 86 (2012)] Small? + 22

23 Hadronic light-by-light: meson exchange Off shell form-factor: Quark propagator Quark-photon vertex Bethe-Salpeter amplitude πγγ form factor Off-shell prescription a μ π,η,η = [Goecke, Fischer and RW, PRD 83 (2011) ] 23

24 Hadronic light-by-light: quark loop Γ μ = λ i L μ μ i + τ i T i i=1,4 i=1,8 Gauge part fixed by WTI Transverse part contains vector mesons ENJL result DSE result L 1 + T 1 : 21 ± [Bijnens, Pallante and Prades, PRL 75 (1995)] L 1 + T 1 : 107 ± [Goecke, Fischer and RW, PRD 87 (2013) ] 24

25 Hadronic light-by-light: quark loop ENJL DSE 25

26 ENJL vs DSE: gauge part Γ μ = λ i L μ μ i + τ i T i i=1,4 i=1,8 L 1 μ = γ μ ENJL/DSE both satisfy the Ward-Takahashi identity λ 1 k 1, k 2 = Z f (k 2 1 ) + 1 Z f (k 2 2 ) ENJL: Z_f = 1 DSE: Z_f < 1 Gauge part roughly cancels Z_f from quarks Enhancement in due to momentum dependent mass function 26

27 ENJL vs DSE: transverse part DSE Couples to vector meson: SUPPRESSION Non-contact interaction: relative momentum; DAMPS suppression ENJL Point-like vector meson: SUPPRESSION Contact interaction: no relative momentum

28 ENJL vs DSE: transverse part DSE Couples to vector meson: SUPPRESSION Non-contact interaction: relative momentum; DAMPS suppression Large contributions from the quark loop arises due to the full momentum dependence that is considered here 28

29 Hadronic light-by-light: results Group Model π π, η, η Quark loop a μ LBL BPP ENJL 59(11) 85(13) 21(3) 83(32) HKS HLS 57(4) 83(6) 10(11) 89(16) KN LMD+V 58(10) 83(11) 83(12) MV LMD+V 77(5) 114(10) 114(10) DB NLχQ 65(2) N LMD+V 72(12) 99(16) 99(16) GR CχQM 68(3) (5) GFW DSE 58(1) 81(2) 107(3) 188(5) BPP: Bijnens et al, PRL 75 (1995) 1447 HKS: Hayakawa et al, PRL 75 (1995) 790 KN: Knecht et al, PRD 65 (2002) MV: Melnikov et al, PRD 70 (2004) DB: Dorokhov et al, PRD 78 (2008) N: Nyffeler, PRD 79 (2009) GR: Greynat et al, JHEP 1207 (2012) 020 GFW: PRD 87 (2013)

30 Overview Introduction Hadronic Vacuum Polarisation Hadronic Light-by-Light Scattering Conclusions 30

31 Conclusions Dynamical ab initio calculation of muon g 2 desirable Model approximations can underestimate contributions Lattice QCD (LO now, NLO future) Functional methods (now, but more work needed) QCD corrections from theory not necessarily robust Possible underestimation of errors (overconfidence?) No statement regarding beyond-the-standard model from muon g 2 Thank you! 31

32 Conclusions: beyond standard model? Anomalous magnetic moment of the muon as the harbinger of new physics? first get QCD under control Experiment Theory Our value a μ = ± [Benett et al, PRD 73 (2006) ] a μ = ± [Hagiwara et al, JPG 38 (2011) ] a μ = ± Error Estimate? Just a guess