Effective field theory estimates of the hadronic contribution to g 2

Transcription

1 Effective field theory estimates of the hadronic contribution to g 2 Fred Jegerlehner fjeger@physik.hu-berlin.de Work in collaboration with Maurice Benayoun et al EPJ C 72 (2012) 1848 and extension RMC WG meeting Mainz, September 2012 Supported by Cooperate Research Center CRC/SFB 1044 of DFG F. Jegerlehner RMC WG Mainz, September 2012

2 Outline of Talk: Effective field theory: the Resonance Lagrangian Approach, broken HLS A minimal version: VMD + sqed solving the τ vs e + e puzzle Global fit of BHLS parameters and prediction of F π (s) SM prediction of a µ using BHLS predictions for a LO,had µ Lessons and Outlook F. Jegerlehner RMC WG Mainz, September

3 Effective field theory: the Resonance Lagrangian Approach Principles to be included: Chiral Structure of QCD, VMD & electromagnetic gauge invariance. General framework: resonance Lagrangian extension of chiral perturbation theory (CHPT), i.e. implement VMD model with Chiral structure of QCD. Specific version Hidden Local Symmetry (HLS) effective Lagrangian CHPT systematic and unambiguous approach to low energy effective QCD: Spontaneously broken chiral symmetry S U(3) S U(3) [pseudoscalars as Nambu-Goldstone bosons] plus expansion in low momenta and chiral SB effects [p m q ; q = u, d, s], limitation: ceases to converge for E 400 MeV, lacks vector resonances ρ, ω,... Vector meson dominance model (effective theory of spin 1 vector resonances ρ, ω,...): photon has direct coupling to ρ 0. In quark model (QCD, SM) photons obviously couple to hadrons via the charged quark F. Jegerlehner RMC WG Mainz, September

4 γ q γ ρ 0 VDM: L q γρ = em2 ρ g ρ ρ µ A µ or = e 2g ρ ρ µν F µν Resonance Lagrangian approach: can be viewed as an attempt to implement the VMD model in a way which is consistent with the chiral structure of QCD. Construction of HLS model: Like in CHPT the basic fields are the unitary matrix fields ξ L,R = exp [ ±i P/ f π ], where P = P 8 + P 0 is the S U(3) matrix of pseudoscalar fields, with P 0 and P 8 the basic singlet and octet fields, respectively. The pseudoscalar field matrix P : P 8 = π η 8 + π + K + π 1 2 π η 8, P 0 = 1 6 diag(η 0, η 0, η 0 ) ; (π 3, η 8, η 0 ) (π 0, η, η ) F. Jegerlehner RMC WG Mainz, September

5 The hidden local symmetry (HLS) ansatz is an extension of the CHPT non-linear sigma model to a non-linear chiral Lagrangian [Tr µ ξ + µ ξ ] based on the symmetry pattern G global /H local, where G = S U(3) L S U(3) R is the chiral group of QCD and H = S U(3) V the vector subgroup. The hidden local S U(3) V requires the spin 1 vector meson fields, represented by the S U(3) matrix field V µ, to be gauge fields. The needed covariant derivative reads D µ = µ i g V µ, allows to include the couplings to the electroweak gauge fields A µ, Z µ and W µ ±. The vector field matrix is given by : V = 1 2 (ρ I + ω I )/ 2 ρ + K + ρ ( ρ I + ω I )/ 2 K 0 K K 0 φ I F. Jegerlehner RMC WG Mainz, September

6 The unbroken HLS Lagrangian is then given by L HLS = L A + L V ; L A/V = f 2 π 4 Tr [L ± R]2 where L = [ D µ ξ L ] ξ + L and R = [ D µ ξ R ] ξ + R. The covariant derivatives are given by : D µ ξ L = µ ξ L igv µ ξ L + iξ L L µ D µ ξ R = µ ξ R igv µ ξ R + iξ R R µ with : L µ = eqa µ + g 2 cos θ W (T z sin 2 θ W )Z µ + g 2 2 (W + µ T + + W µ T ) R µ = eqa µ g 2 cos θ W sin 2 θ W Z µ F. Jegerlehner RMC WG Mainz, September

7 In fact the global chiral symmetry G global is well known not to be realized as an exact symmetry in nature which implies that the ideal HLS symmetry evidently is not a symmetry of nature either. It evidently has to be broken appropriately in order to provide a realistic low energy effective theory mimicking low energy effective QCD. Corresponding to the strength of the breaking, usually, this has be done in two steps, breaking of S U(3) in a first step and breaking S U(2) isospin in a second step. Unlike in CHPT (perturbed non-linear σ model) where one is performing a systematic low energy expansion, expanding in low momenta and the quark masses, here we introduce symmetry breaking as phenomenological parameters to be fixed from appropriate data, since a systematic low energy expansion a lá CHPT ceases to converge at energies above about 400 MeV, while we attempt to model phenomenology up to including the φ resonance. F. Jegerlehner RMC WG Mainz, September

8 The broken HLS Lagrangian (BHLS) is then given by L BHLS = L A + L V + L thooft ; L A/V = f 2 π 4 Tr { [L ± R] X A/V } 2 with 6 phenomenological chiral SB parameters. SB pattern: X I = diag(q I, y I, z I ) ; q I 1, y I 1 z I 1 ; I = V, A. Anomalous sector as well: simplest Wess-Zumino-Witten Lagrangian N c L WZW = α π 12F π π η η 0 F µν F µν, which is reproducing the ABJ anomaly and is responsible for π 0 γγ etc. F. Jegerlehner RMC WG Mainz, September

9 A minimal version: VMD + sqed solving the τ vs e + e puzzle Effective Lagrangian L = L γρ + L π L π = D µ π + D +µ π m 2 ππ + π ; D µ = µ i e A µ i g ρππ ρ µ L γρ = 1 4 F µν F µν 1 4 ρ µν ρ µν + M2 ρ 2 ρ µ ρ µ + e 2 g ρ ρ µν F µν F. Jegerlehner RMC WG Mainz, September

10 Self-energies: pion loops to photon-rho vacuum polarization µν (π) iπ γγ (q) = +. µν (π) iπ γρ (q) = +. µν (π) iπ ρρ (q) = +. Irreducible self-energy contribution at one-loop bare γ ρ transverse self-energy functions Π γγ = e 2 48π 2 f (q2 ), Π γρ = eg ρππ 48π 2 f (q2 ) and Π ρρ = g2 ρππ 48π 2 f (q2 ), F. Jegerlehner RMC WG Mainz, September

11 µν (π) iπ γγ (q) = +. µν (π) iπ γρ (q) =. µν (π) iπ ρρ (q) = +. Previous calculations, consider mixing term to be constant The e + e π + π matrix element in sqed is given by M = i e 2 vγ µ u (p 1 p 2 ) µ F π (q 2 ) with F π (q 2 ) = 1. In our extended VMD model we have the four terms F. Jegerlehner RMC WG Mainz, September

12 e + γ π + γ ρ ρ γ ρ e π Diagrams contributing to the process e + e π + π. F π (s) e 2 D γγ + eg ρππ D γρ g ρee ed ργ g ρee g ρππ D ρρ, Properly normalized (VP subtraction: e 2 (s) e 2 ): F π (s) = [ e 2 D γγ + e (g ρππ g ρee ) D γρ g ρee g ρππ D ρρ ] / [ e 2 D γγ ] Typical couplings g ρππ bare = , g ρππ ren = , g ρee = , x = g ρππ /g ρ = g ρππ = 48 π Γ ρ /(β 3 ρ M ρ ) ; g ρee = 12π Γ ρee /M ρ F. Jegerlehner RMC WG Mainz, September

13 Good old idea: use isospin symmetry to include existing high quality τ data (including isospin corrections) e + e τ ν µ γ W γ ū, d u, d isospin rotation W ū d π + π, [I = 1] π 0 π, Corrected data: large discrepancy [ 10%] persists! τ vs. e + e problem! [manifest since 2002] F. Jegerlehner RMC WG Mainz, September

14 Recent: τ (charged channel) vs. e + e (neutral channel) puzzle resolved F.J.& R. Szafron, ρ γ interference (absent in charged channel): µν (π) iπ γρ (q) = +. 0 (s) = r ργ (s) R IB (s) (s) τ require to be corrected for missing ρ γ mixing! results obtained from e + e data is what goes into a µ F. Jegerlehner RMC WG Mainz, September

15 τ decays e + e +CVC ALEPH 1997 ALEPH 2005 OPAL 1999 CLEO 2000 Belle 2008 τ combined CMD SND 2006 KLOE 2008 KLOE 2010 BABAR 2009 e + e combined ± 2.65 ± ± 4.00 ± ± 7.27 ± ± 4.11 ± ± 0.53 ± ± 1.42 ± ± 2.76 ± ± 1.40 ± ± 0.34 ± ± 0.71 ± ± 0.37 ± ± 0.87 ± 2.18 a µ [ππ], I = 1, ( ) GeV I=1 part of a had µ [ππ] F. Jegerlehner RMC WG Mainz, September

16 τ decays e + e +CVC ALEPH 1997 ALEPH 2005 OPAL 1999 CLEO 2000 Belle 2008 τ combined CMD SND 2006 KLOE 2008 KLOE 2010 BABAR 2009 e + e combined ± 2.65 ± ± 4.00 ± ± 7.27 ± ± 4.11 ± ± 0.53 ± ± 1.40 ± ± 2.76 ± ± 1.40 ± ± 0.34 ± ± 0.71 ± ± 0.37 ± ± 0.87 ± 2.18 a µ [ππ], I = 1, ( ) GeV I=1 part of a had µ [ππ] F. Jegerlehner RMC WG Mainz, September

17 F π (E) 2 in units of e + e I=1 (CMD-2 GS fit) Best proof : F. Jegerlehner RMC WG Mainz, September

18 F π (E) 2 in units of e + e I=1 (CMD-2 GS fit) Best proof : F. Jegerlehner RMC WG Mainz, September

19 Is our model viable? How photons couple to pions? This is obviously probed in reactions like γγ π + π, π 0 π 0. Data infer that below about 1 GeV photons couple to pions as point-like objects (i.e. to the charged ones overwhelmingly), at higher energies the photons see the quarks exclusively and form the prominent tensor resonance f 2 (1270). The π 0 π 0 cross section in this figure is enhanced by the isospin symmetry factor 2, by which it is reduced in reality. F. Jegerlehner RMC WG Mainz, September

20 Normalization as in reality: σ(π 0 π 0 )/σ(π + π ) = 1 2 at the f 2(1270) peak (art work by Mike Pennington) F. Jegerlehner RMC WG Mainz, September

21 γ γ π + π+,π0 π π,π 0 ū, d u,d Di-pion production in γγ fusion. At low energy we have direct π + π production and by strong rescattering π + π π 0 π 0, however with very much suppressed rate. Above about 1 GeV, resolved q q couplings seen. Strong tensor meson resonance in ππ channel f 2 (1270) with photons directly probe the quarks! Photons seem to see pions below 1 GeV Photons definitely look at the quarks in f 2 (1270) resonance region We apply the sqed model up to GeV (relevant for a µ ). This should be pretty save (still we assume a 10% model uncertainty) Switching off the electromagnetic interaction of pions, is definitely not a realistic approximation in trying to describe what data we see in e + e π + π F. Jegerlehner RMC WG Mainz, September

22 New BaBar ISR 2π result: arxiv: v1 10 May 2012 e + e ee BaBar ee KLOE ee CMD-2 ee SND τ τ ALEPH τ CLEO τ OPAL τ Belle a 2π,LO µ γ ρ correction: not applied F. Jegerlehner RMC WG Mainz, September

23 New BaBar ISR 2π result: arxiv: v1 10 May 2012 e + e ee BaBar ee KLOE ee CMD-2 ee SND τ τ ALEPH τ CLEO τ OPAL τ Belle a 2π,LO µ γ ρ correction: applied F. Jegerlehner RMC WG Mainz, September

24 a 2π,LO µ [2m π 1.8 GeV]: (514.1 ± 3.8) [BaBar] (507.8 ± 3.2) [ee all] (508.7 ± 2.5) [ee + τ JS] a the µ : (54) [BaBar] (50) [ee all] (46) [ee + τ JS] δa µ = a exp µ a the µ : (224 ± 83) 10 11, 2.7 σ [BaBar] (287 ± 80) 10 11, 3.6 σ, [ee all] (278 ± 78) 10 11, 3.6 σ, [ee + τ JS] F. Jegerlehner RMC WG Mainz, September

25 Global fit of BHLS parameters and prediction of F π (s) Fit τ + IB from PDG vs π + π F. Jegerlehner RMC WG Mainz, September

26 Data below E 0 = 1.05 GeV (just above the φ) constrain effective Lagrangian couplings, using 45 different data sets (6 annihilation channels and 10 partial width decays). Effective theory predicts cross sections: π + π, π 0 γ, ηγ, η γ, π 0 π + π, K + K, K 0 K 0 (83.4%), Missing part: 4π, 5π, 6π, ηππ, ωπ and regime E > E 0 evaluated using data directly and pqcd for perturbative region and tail Including self-energy effects is mandatory (γρ-mixing, ρω-mixing..., decays with proper phase space, energy dependent width etc) Method works in reducing uncertainties by using indirect constraints Able to reveal inconsistencies in data. In our case in region [1.00,1.05] GeV tension between KK and 3π data sets. All data: Solution A 71.2% CL; excluding 3π above 1 GeV: Solution B 97.0% CL. Conflict in data?, model? F. Jegerlehner RMC WG Mainz, September

27 Singling out effective resonance Lagrangian by global fit is expected to help in improving EFT calculations of hadronic light-by-light scattering (such concept so far missing) F. Jegerlehner RMC WG Mainz, September

28 Fit of τ +PDG vs π + π data F. Jegerlehner RMC WG Mainz, September

29 Fit of BaBar vs other π + π data F. Jegerlehner RMC WG Mainz, September

30 Best fits of τ +PDG + selected π + π data F. Jegerlehner RMC WG Mainz, September

31 SM prediction of a µ using BHLS predictions for a LO,had µ a µ Values (incl. τ) Values (excl. τ) no ISR with ISR [new] scan + ISR scan only LO hadronic ± ± ± ± 4.99 HO hadronic 9.97 ± 0.09 LBL 10.5 ± 2.6 QED ± EW ± 0.10 had ± 0.03 Higgs,top,3 loop Theor ± ± ± ± 5.55 Exper ± 6.3 a µ ± ± ± ± 8.40 S.D. (nσ) 4.51σ 4.46σ 4.32σ 4.63σ The various contributions to a µ. a µ = (a µ ) exp (a µ ) the is given in units of F. Jegerlehner RMC WG Mainz, September

32 Hadronic VP contributions to a µ with FSR corrections included. Final State Range (GeV) Contribution (incl. τ) Contribution (excl. τ) Solution A Solution B Solution A Solution B e + e hadrons threshold [1.90] [2.10] [2.15] [2.27] missing channels threshold (0.40)(0.40)[0.57] J/ψ 8.51(0.40)(0.38)[0.55] Υ 0.10(0.00)(0.10)[0.10] hadronic (1.05, 2.00) 60.76(0.22)(3.93)[3.94] hadronic (2.00, 3.10) 21.63(0.12)(0.92)[0.93] hadronic (3.10, 3.60) 3.77(0.03)(0.10)[0.10] hadronic (3.60, 5.20) 7.64(0.04)(0.05)[0.06] pqcd (5.20, 9.46) 6.19(0.00)(0.00)[0.00] hadronic (9.46, 13.00) 1.28(0.01)(0.07)[0.07] pqcd (13.00, ) 1.53(0.00)(0.00)[0.00] Total ± 4.13 tot + missing channels Total Model threshold ± ± ± ± 4.71 F. Jegerlehner RMC WG Mainz, September

33 LO had Our favored evaluation is : aµ = ( ± 4.52) a the µ = ( ± 5.22) 10 10, a µ = a exp µ a the µ = (39.21 ± 5.22 the ± 6.3 exp ) 10 10, The associated fit probability is 94% and the significance for a µ is 4.8σ LO had Including all data: aµ = ( ± 4.23) a the µ = ( ± 4.97) 10 10, a µ = a exp µ a the µ = (35.51 ± 4.97 the ± 6.3 exp ) 10 10, The associated fit probability is 76% and the significance for a µ is 4.4σ (requires rewighting of e + e π + π π 0 in vicinity of the φ as well as KLOE08 and BaBar e + e π + π ) F. Jegerlehner RMC WG Mainz, September

34 Comparison of various Fits F. Jegerlehner RMC WG Mainz, September

35 Comparison with other Results F. Jegerlehner RMC WG Mainz, September

36 Lessons and Outlook Effective field theory is the only way to understand relationships between different channels, like e + e annihilation cross-sections and τ decay spectra Global fit strategies allow to single out variants of effective resonance Lagrangian models Models for individual channels can parametrize data, but do not allow to understand them and their relation to other channels We get perfect fits for F π (s) 2 up to just above the φ without higher ρ s ρ, ρ, which seem to be mandatory in Gounaris-Sakurai type fits. τ data in our approach play special role, much simpler than e + e with its intricate γ ρ 0 ω φ mixing. π + π cross-section from τ spectra plus isospin breaking encoded in ρ e + e [main issue in τ vs e + e ], ω π + π etc F. Jegerlehner RMC WG Mainz, September

37 RLA type analyses provide analytic shapes for amplitudes, and such physical shape information is favorable over ad hoc data interpolations (the simplest being the trapezoidal rule, which is known to be problematic when data are sparse or strongly energy dependent). Limitations: large couplings, higher order effects?, range of validity, e.g. point-like pions (sqed) etc. Our analysis is a starting point to be confronted with other RLA versions and implementations including higher order effects. Good starting point for reevaluation of hadronic light-by-light contribution to muon g 2. So far no globally verified framework applied. Last but not least: there is no τ vs e + e problem F. Jegerlehner RMC WG Mainz, September

38 And here we are: BNL CERN III CERN II CERN I th QED 6th 8th hadronic VP hadronic LBL weak New Physics??? SM precision a µ uncertainty [ppm] Sensitivity of g 2 experiments to various contributions. The increase in precision with the BNL g 2 experiment is shown as a cyan vertical band. New Physics is illustrated by the deviation (a exp µ a the µ )/a exp µ F. Jegerlehner RMC WG Mainz, September

39 Thank you for your attention! F. Jegerlehner RMC WG Mainz, September