Leading-order hadronic contribution to the anomalous magnetic moment of the muon from N f = twisted mass fermions

Transcription

1 Leading-order hadronic contribution to the anomalous magnetic moment of the muon from N f = twisted mass fermions Grit Hotzel 1 in collaboration with Florian Burger 1, Xu Feng 2, Karl Jansen 3, Marcus Petschlies 4, Dru B. Renner 5 1 Humboldt University Berlin, Germany 2 KEK, Japan 3 NIC, DESY Zeuthen, Germany 4 The Cyprus institute, Cyprus 5 Jefferson Lab, USA Lattice 213, Mainz, Germany Grit Hotzel (HU Berlin) The muon g 2 2/8/13 1 / 18

2 Why the muon s anomalous magnetic moment? The anomalous magnetic moment of the muon, a, can be measured very precisely: [B. Lee Roberts, Chinese Phys. C 34, 21] a exp = (63) 1 11 a SM = (49) 1 11 [Hagiwara et al.,j. Phys. G38, 211] HMNT (6) JN (9) Davier et al, τ (1) Davier et al, e + e (1) JS (11) HLMNT (1) HLMNT (11) experiment BNL BNL (new from shift in λ) There is a 3σ discrepancy between a exp and a SM : a exp a SM = 261(8) 1 11 a Grit Hotzel (HU Berlin) The muon g 2 2/8/13 2 / 18

3 Charm quark necessary to reach required precision Current discrepancy [Hagiwara et al.,j. Phys. G38, 211] a exp a SM = 261(8) 1 11 charm quark contribution computed in perturbation theory [Bodenstein, Dominguez, Schilcher, Phys.Rev. D85, 212] a hvp,c = 144(1) 1 11 comparable to hadronic light-by-light scattering contribution [Prades, de Rafael, Vainshtein, arxiv:91.36 [hep-ph], 29] a hlbl = 15(26) 1 11 and also to electroweak contribution [Jegerlehner, Nyffeler, Phys. Rept. 477, 29] a EW = 153(2) 1 11 Grit Hotzel (HU Berlin) The muon g 2 2/8/13 3 / 18

4 Leading hadronic contribution a hvp a QCD = a lo,hvp + a ho,hvp + a lbl γ can be computed directly in Euclidean space-time [T. Blum, PRL 91, 23] a hvp = α 2 had where Π R (Q 2 ) = Π(Q 2 ) Π() main ingredient: hadronic vacuum polarisation tensor Π ν (Q) = d 4 x e iq (x y) J em (x)jν em (y) = (Q Q ν Q 2 g ν )Π(Q 2 ) dq 2 Q 2 w ( Q 2 m 2 ) Π R (Q 2 ) with J em (x) = 2 3 u(x)γ u(x) 1 3 d(x)γ d(x)+ 2 3 c(x)γ c(x) 1 3 s(x)γ s(x) Grit Hotzel (HU Berlin) The muon g 2 2/8/13 4 / 18

5 Mixed-action set-up configurations generated by ETMC [Baron et al., JHEP 16, 21] light quarks: twisted mass action for mass-degenerate fermion doublet [Frezzotti, Rossi, JHEP 48, 24] S F [χ, χ, U] = x χ(x) [ D W + m + i q γ 5 τ 3] χ(x) heavy sea quarks: twisted mass action for non-degenerate fermion doublet [Frezzotti, Rossi, Nucl. Phys. Proc. Suppl. 128, 24] S F [χ h, χ h, U] = [ χ h (x) D W + m + i σ γ 5 τ 1 + δ τ 3] χ h (x) x heavy valence quarks: Osterwalder-Seiler action [Frezzotti, Rossi, JHEP 41, 24] S F [χ h, χ h, U] = x [ ( c χ h (x) D W + m + i s ) γ 5 ] χ h (x) tune bare mass parameters c/s such that physical kaon and D-meson masses are reproduced Grit Hotzel (HU Berlin) The muon g 2 2/8/13 5 / 18

6 The N f = ensembles Ensemble β a[fm] L 3 T m PS [MeV] L[fm] D D D45.32sc B25.32t B B B B B A A A Grit Hotzel (HU Berlin) The muon g 2 2/8/13 6 / 18

7 First N f = 2 configurations at the physical point More details: Talk by Bartosz Kostrzewa, Monday, 16:5 again use Iwasaki action in gauge sector add clover-term to twisted mass action for non-degenerate fermion doublet S F [χ, χ, U] = [ χ(x) D W + m + i q γ 5 τ 3] χ(x) x [ ] i + c SW χ(x) 4 σ νf ν χ(x) very preliminary parameters of first ensemble: β c SW a[fm] L 3 T m PS [MeV] L[fm] x Grit Hotzel (HU Berlin) The muon g 2 2/8/13 7 / 18

8 How the observables are determined use conserved (point-split) vector current J C (x) = 1 ( χ(x + ˆ)(1 + γ )U 2 (x)q el χ(x) ) χ(x)(1 γ )U (x)q el χ(x + ˆ) where Q el = diag( 2 3, 1 3 ) use redefinition [Feng, Jansen, Petschlies, Renner, PRL 17, 211] ) a hvp = α 2 dq 2 Q 2 w ( Q 2 H 2 H 2 phys m 2 Π R (Q 2 ) which goes to a hvp for m PS m π, i.e. when H H phys effectively, redefinition of muon mass m = m H H phys in the following will always use H = m V - ρ-meson mass Grit Hotzel (HU Berlin) The muon g 2 2/8/13 8 / 18

9 Fitting the hadronic vacuum polarisation function have Π( ˆQ 2 ) depending on discrete momenta to obtain smooth function fit this for each flavour to Π(Q 2 ) = (1 θ(q 2 Q 2 match))π low (Q 2 ) + θ(q 2 Q 2 match)π high (Q 2 ) with and Π low (Q 2 ) = Π high (Q 2 ) = C 1 k= M i=1 g 2 i c k (Q 2 ) k + m 2 i Q 2 + m 2 i ( B 1 a j (Q 2 ) j N 1 + j= ) b l (Q 2 ) l log(q 2 ) different matching conditions and functions possible Padé approximants [Aubin, Blum, Golterman, Peris, Phys.Rev. D86, 212] under investigation Grit Hotzel (HU Berlin) The muon g 2 2/8/13 9 / 18 l=

10 Light quark contribution on N f = sea 6e-8 5.5e-8 5e-8 4.5e-8 H = m V a ud 4e-8 3.5e-8 3e-8 2.5e-8 2e-8.5 N f = 2 result a =.86 fm, L = 2.8 fm a =.78 fm, L = 1.9 fm a =.78 fm, L = 2.5 fm a =.78 fm, L = 3.7 fm a =.61 fm, L = 1.9 fm a =.61 fm, L = 2.9 fm.1.15 m 2 [ PS GeV 2 ] N f = result: a hvp,ud = 5.67(11) 1 8 N f = 2 result: a hvp,ud = 5.72(16) 1 8 [Feng, Jansen, Petschlies, Renner, PRL 17, 211].2.25 Grit Hotzel (HU Berlin) The muon g 2 2/8/13 1 / 18

11 Light quark contribution on N f = sea Comparing to preliminary N f = 2 result at physical pion mass 6e-8 5e-8 H = m V a ud 4e-8 3e-8 2e-8 1e-8 Preliminary.5 N f = 2 result a =.86 fm, L = 2.8 fm a =.78 fm, L = 1.9 fm a =.78 fm, L = 2.5 fm a =.78 fm, L = 3.7 fm a =.61 fm, L = 1.9 fm a =.61 fm, L = 2.9 fm.1.15 m 2 [ PS GeV 2 ].2.25 H = 1 N f = result: a hvp,ud = 5.67(11) 1 8 new N f = 2 result at physical point: a hvp,ud = 5.55(7) 1 8 Grit Hotzel (HU Berlin) The muon g 2 2/8/13 11 / 18

12 Adding the strange quark in the valence sector for strange quark lattice artefacts have to be taken into account a strange 5.6e-9 5.4e-9 5.2e-9 5e-9 4.8e-9 4.6e-9 4.4e-9 data extrapolated to mπ CL with linear fit light sector: cannot discriminate a 2 effects 6.5e-8 data extrapolated to mπ CL with constant fit CL with linear fit in a 2 4.2e-9 4e-9 6e-8 3.8e a 2 [ fm 2] for combined chiral and continuum extrapolation will use a ud 5.5e-8 5e-8 4.5e a 2 [ fm 2] a hvp,uds (m PS, a) = A + B m 2 PS + C a 2 with fit parameters A, B, C Grit Hotzel (HU Berlin) The muon g 2 2/8/13 12 / 18

13 Three-flavour contribution on N f = sea 7e-8 6e-8 H = m V a uds 5e-8 4e-8 3e-8 2e-8.5 a result dispersive analysis & pheno a =.61 fm, L = 1.9 fm a =.61 fm, L = 2.9 fm a =.78 fm, L = 1.9 fm a =.78 fm, L = 2.5 fm a =.78 fm, L = 3.7 fm a =.86 fm, L = 2.8 fm a (m PS,.61 fm) a (m PS,.78 fm) a (m PS,.86 fm).1 m 2 PS.15 [ GeV 2 ] N f = result: a hvp,uds = 6.55(21) 1 8 three-flavour result extracted from dispersive analysis of [Jegerlehner, = 6.79(5) 1 8 Szafron, Eur. Phys. J C71, 211]: a hvp,uds Grit Hotzel (HU Berlin) The muon g 2 2/8/13 13 /

14 The four-flavour contribution on N f = sea 7e-8 6.5e-8 6e-8 H = m V a udsc 5.5e-8 5e-8 4.5e-8 4e-8 3.5e-8 3e-8.5 a result dispersive analysis a =.61 fm, L = 1.9 fm a =.61 fm, L = 2.9 fm a =.78 fm, L = 1.9 fm a =.78 fm, L = 2.5 fm a =.78 fm, L = 3.7 fm a =.86 fm, L = 2.8 fm a (m PS,.61 fm) a (m PS,.78 fm) a (m PS,.86 fm).1.15 m 2 [ PS GeV 2 ] N f = result: a hvp = 6.74(21) 1 8 result from dispersive analysis: [Jegerlehner, Szafron, Eur. Phys. J C71, 211] a hvp = 6.91(5) Grit Hotzel (HU Berlin) The muon g 2 2/8/13 14 / 18

15 Systematic uncertainties from choosing fit ranges for vector mesons: V = from choosing different number of terms in fit function: MNBC = found to be negligible: m PS L > 3.8, m PS < 4 MeV, varying matching momentum between [1 GeV 2, 3 GeV 2 ] not quantified yet: disconnected contributions, wrong sea quark masses preliminary final result: a hvp = 6.74(21)(18) 1 8 Grit Hotzel (HU Berlin) The muon g 2 2/8/13 15 / 18

16 Comparison of results for a hvp a hvp in 1 8 for different numbers of valence quarks: u,d u, d, s u, d, s, c this work 5.67(11) 6.55(21) 6.74(27) ETMC (16) - - Mainz (66) 6.18(64) - RBC-UKQCD (33) - HLS estimate 211 (dispersive analysis + flavour weighting) 5.59(4) 6.71(5) 6.83(5) ETMC 211: [Feng, Jansen, Petschlies, Renner, PRL 17, 211] Mainz 211: [Della Morte, Jäger, Jüttner, Wittig, JHEP 123, 212] RBC-UKQCD 211: [Boyle, Del Debbio, Kerrane, Zanotti, Phys. Rev. D85, 212] HLS estimate 211: [Benayoun, David, DelBuono, Jegerlehner, Eur. Phys. J. C72, 212] Grit Hotzel (HU Berlin) The muon g 2 2/8/13 16 / 18

17 Summary First N f = lattice calculation of a hvp gives compatible result with dispersive analyses. Davier et al. (τ) Davier et al. (e + e ) Jegerlehner, Szafron Hagiwara et al. HLS estimate This work 6.6e-8 6.8e-8 7e-8 a hvp 7.2e-8 7.4e-8 Modified method [Feng, Jansen, Petschlies, Renner, PRL 17, 211] works for N f = computation. Chiral extrapolation in light sector to be checked with computation at physical pion mass. Grit Hotzel (HU Berlin) The muon g 2 2/8/13 17 / 18

18 Outlook improve data by more statistics, especially in heavy sector, and all-mode-averaging use Padé approximants disconnected contributions more N f = 2 configurations at physical point N f = configurations at physical point, probably several lattice spacings needed include isospin breaking effects different observables: α hvp QED, Adler function, weak mixing angle, S-parameter light-by-light scattering Grit Hotzel (HU Berlin) The muon g 2 2/8/13 18 / 18

19 Why it works redefinition [Feng, Jansen, Petschlies, Renner, PRL 17, 211] a hvp = α 2 dq 2 Q 2 w ( Q 2 H 2 H 2 phys m 2 ) Π R (Q 2 ) which goes to a hvp for m PS m π, i.e. when H H phys effectively, redefinition of muon mass m = m H H phys leading vector meson contribution a hvp α 2 gv 2 m 2 mv 2 strong dependence on m PS via m V m V [GeV] a=.79 fm L=1.6 fm a=.79 fm L=1.9 fm a=.79 fm L=2.5 fm a=.63 fm L=1.5 fm a=.63 fm L=2. fm model PDG 2 m PS [GeV 2 ] Grit Hotzel (HU Berlin) The muon g 2 2/8/13 19 / 18

20 Comparison with Padé approximants Padé approximants model-independent, systematically improvable way of fitting vacuum polarisation [Aubin et al., Phys.Rev. D86, 212] vacuum polarisation integrated up to Q 2 max = 1.5 GeV 2 : 7e-8 6e-8 5e-8 H = m V a ud 4e-8 3e-8 2e-8 H = 1 1e m 2 [ PS GeV 2 ].1 M1N2 [1, 1] Padé will use Padé approximants when chiral extrapolation no longer needed Grit Hotzel (HU Berlin) The muon g 2 2/8/13 2 /

21 Example for standard fit β=1.95, L/a=32, light = Π light fit data a 2 Q 2 Grit Hotzel (HU Berlin) The muon g 2 2/8/13 21 / 18