Hadronic light-by-light scattering in the muon g 2

Transcription

1 Hadronic light-by-light scattering in the muon g 2 Andreas Nyffeler Institut für Kernphysik Johannes Gutenberg Universität Mainz, Germany nyffeler@kph.uni-mainz.de Institut für Kern- und Teilchenphysik Technische Universität Dresden, Germany October 30, 2014

2 Outline Part I: Status of the muon g 2 Basics of the anomalous magnetic moment Theory of the muon g 2: QED, weak interactions, hadronic contributions Part II: Hadronic light-by-light scattering in the muon g 2 Starting point, quark-gluon versus hadronic picture, model-based approaches Main results Further partial results for single-resonance poles and exchanges, potential problems with pion-loop and quark-loop New development: data driven approach using dispersion relations Conclusions

3 Part I: Status of the muon g 2

4 Basics of the anomalous magnetic moment Electrostatic properties of charged particles: Charge Q, Magnetic moment µ, Electric dipole moment d For a spin 1/2 particle: µ = g e 2m s, g {z = 2 (1 + a), a = 1 (g 2) : anomalous magnetic moment } 2 Dirac Long interplay between experiment and theory: structure of fundamental forces In Quantum Field Theory (with C,P invariance): γ(k) p p 2 = ( ie)ū(p 6 ) 4γ µ F 1(k 2 ) + iσµν k ν {z } 2m F2(k 2 7 ) 5 u(p) {z } Dirac Pauli 3 F 1(0) = 1 and F 2(0) = a a e: Test of QED. Most precise determination of α = e 2 /4π. a µ: Less precisely measured than a e, but all sectors of Standard Model (SM), i.e. QED, Weak and QCD (hadronic), contribute significantly. Sensitive to possible contributions from New Physics. Often (but not always!): «2 «2 ml mµ a l more sensitive than a e [exp. precision factor 19] m NP m e

5 Tests of the Standard Model and search for New Physics Search for New Physics with two complementary approaches: 1 High Energy Physics: e.g. Large Hadron Collider (LHC) at CERN Direct production of new particles e.g. heavy Z resonance peak in invariant mass distribution of µ + µ at M Z. p p q q Z µ + µ 2 Precision physics: e.g. anomalous magnetic moments a e, a µ Indirect effects of virtual particles in quantum corrections Deviations from precise predictions in Standard Model «2 ml For M Z m l : a l M Z No decoupling for new light vector mesons, e.g. dark photon with M V (10 100) MeV. a e, a µ allow to exclude some models of New Physics or to constrain their parameter space. γ Z µ µ

6 Milestones in measurements of a µ Authors Lab Muon Anomaly Garwin et al. 60 CERN (14) Charpak et al. 61 CERN (22) Charpak et al. 62 CERN (5) Farley et al. 66 CERN (3) Bailey et al. 68 CERN (31) Bailey et al. 79 CERN (84) Brown et al. 00 BNL (59) (µ + ) Brown et al. 01 BNL (14)(6) (µ + ) Bennett et al. 02 BNL (7)(5) (µ + ) Bennett et al. 04 BNL (8)(3) (µ ) World average experimental value (g 2 Collaboration at BNL, Bennett et al CODATA 2008 value for λ = µ µ/µ p, ratio of muon to proton magnetic moment from muonium hyperfine splitting): a exp µ = ( ± 63) [0.5ppm] Goal of new planned g 2 experiments at Fermilab (partly recycled from BNL: moved ring magnet! Pictures at: ) and J-PARC (completely new concept, not magic γ): δa µ = Theory needs to match this precision! Experimental value for electron a e (Hanneke et al. 08): a exp e = ( ± 0.28) [0.24ppb]

7 Some theoretical comments Anomalous magnetic moments are dimensionless To lowest order in perturbation theory in quantum electrodynamics (QED): = a e = a µ = α 2π [Schwinger 1947/48] Loops with different masses a e a µ - Internal large masses decouple: " «1 2 me = + O m4 e 45 m µ mµ 4 - Internal small masses give rise to large log s of mass ratios: ln mµ m e!# α 2 π» 1 mµ = ln 25 «3 m e 36 + O me α 2 m µ π

8 Muon g 2: Theory In Standard Model (SM): aµ SM = aµ QED + aµ weak + aµ had In contrast to a e, here now the contributions from weak and strong interactions (hadrons) are relevant, since a µ (m µ/m) 2. QED contributions Diagrams with internal electron loops are enhanced. At 2-loops: vacuum polarization from electron loops enhanced by QED short-distance logarithm At 3-loops: light-by-light scattering from electron loops enhanced by QED infrared logarithm [Aldins et al. 69, 70; Laporta + Remiddi 93] e +... a (3) µ lbl» 2 = 3 π2 ln mµ α 3 α = m e π π µ Loops with tau s suppressed (decoupling)

9 QED result up to 5 loops Include contributions from all leptons (Aoyama et al. 2012): α α 2 aµ QED = (17) π {z} π m µ/m e,τ α 3 α (32) (63) {z} π {z} π m µ/m e,τ num. int. α (1.04) {z } π num. int. = (9) {z} (19) {z} (7) {z} (29) {z} m µ/m e,τ c 4 c 5 α(a e ) [36] Earlier evaluation of 5-loop contribution yielded c 5 = 662(20) (Kinoshita, Nio 2006, numerical evaluation of 2958 diagrams, known or likely to be enhanced). New value is 4.5σ from this leading log estimate and 20 times more precise. Aoyama et al. 2012: What about the 6-loop term? Leading contribution from light-by-light scattering with electron loop and insertions of vacuum-polarization loops of electrons into each photon line aµ QED (6-loops)

10 Contributions from weak interaction 1-loop contributions [Jackiw, Weinberg, 1972;... ]: weak, (1) aµ (W ) = weak, (1) aµ (Z) = γ a) b) c) W W µ ν µ Z H 2Gµm µ π O(m2 µ /M2 W ) = (0) Gµm µ 2 ( 1 + 4s 2 W )2 5 16π 2 3 weak, (1) + O(mµ/M 2 Z 2 ) = (2) Contribution from Higgs negligible: aµ (H) for m H 114 GeV. weak, (1) aµ = ( ± 0.02) loop contributions (1678 diagrams): weak, (2) aµ = ( ± 1.80) 10 11, large since G F mµ 2 α π ln M Z Under control. Uncertainties from m H ( GeV), 3-loop (RG): ± , small hadronic uncertainty: ± Total weak contribution: a weak µ = (153.2 ± 1.8) Recent reanalysis by Gnendiger et al 13, using the now known Higgs mass, yields a weak µ = (153.6 ± 1.0) (error: 3-loop, hadronic). m µ

11 Hadronic contributions to the muon g 2: largest source of error QCD: quarks bound by strong gluonic interactions into hadronic states In particular for the light quarks u, d, s cannot use perturbation theory! Possible approaches to QCD: Lattice QCD: often still limited precision Effective quantum field theories with hadrons (ChPT): limited validity Simplifying hadronic models: model uncertainties not controllable Dispersion relations: extend validity of EFT s, reduce model dependence, often not all the needed input data available

12 Hadronic contributions to the muon g 2: largest source of error QCD: quarks bound by strong gluonic interactions into hadronic states In particular for the light quarks u, d, s cannot use perturbation theory! Possible approaches to QCD: Lattice QCD: often still limited precision Effective quantum field theories with hadrons (ChPT): limited validity Simplifying hadronic models: model uncertainties not controllable Dispersion relations: extend validity of EFT s, reduce model dependence, often not all the needed input data available Different types of contributions to g 2: (a) (b) (c) γ (a) Hadronic vacuum polarization (HVP) O(α 2 ), O(α 3 ) (b) Hadronic light-by-light scattering (HLbL) O(α 3 ) (c) 2-loop electroweak contributions O(αG F m 2 µ) 2-Loop EW Small hadronic uncertainty from triangle diagrams. Anomaly cancellation within each generation! Cannot separate leptons and quarks! Z Light quark loop not well defined Hadronic blob e u,d γ Z γ Z µ µ

13 Hadronic vacuum polarization a HVP µ = µ µ Optical theorem (from unitarity; conservation of probability) for hadronic contribution dispersion relation: 2 Im σ(e + e γ hadrons) aµ HVP = 1 α 2 Z 3 π 0 ds s K(s) R(s), R(s)= σ(e+ e γ hadrons) σ(e + e γ µ + µ ) [Bouchiat, Michel 61; Durand 62; Brodsky, de Rafael 68; Gourdin, de Rafael 69] K(s) slowly varying, positive function a HVP µ positive. Data for hadronic cross section σ at low center-of-mass energies s important due to factor 1/s: 70% from ππ [ρ(770)] channel, 90% from energy region below 1.8 GeV. Other method instead of energy scan: Radiative return at colliders with fixed center-of-mass energy (DAΦNE, B- Factories, BES) [Binner, Kühn, Melnikov 99; Czyż et al ] e e + γ γ Hadrons

14 Measured hadronic cross-section Pion form factor F π(e) 2 (ππ-channel) R(s) = 1 4 «3 1 4m2 2 π Fπ(s) 2 s (4m 2 π < s < 9m2 π ) R-ratio: Jegerlehner + Nyffeler 09

15 Hadronic vacuum polarization: some recent evaluations Authors Jegerlehner 08; JN 09 (e + e ) ± 52.6 Contribution to a HVP µ 1011 Davier et al. 09 (e + e ) [+ τ] 6955 ± 41 [7053 ± 45] Teubner et al. 09 (e + e ) 6894 ± 40 Davier et al. 10 (e + e ) [+ τ] 6923 ± 42 [7015 ± 47] Jegerlehner + Szafron 11 (e + e ) [+ τ] ± 47.2 [ ± 46.5 ] Hagiwara et al. 11 (e + e ) ± 42.7 Benayoun at al. 12 (e + e + τ: HLS improved) ± 46.3 Precision: < 1%. Non-trivial because of radiative corrections (radiated photons). Even if values for a HVP µ after integration agree quite well, the systematic differences of a few % in the shape of the spectral functions from different experiments (BABAR, CMD-2, KLOE, SND) indicate that we do not yet have a complete understanding. Use of τ data: additional sources of isospin violation? Ghozzi + Jegerlehner 04; Benayoun et al. 08, 09; Wolfe + Maltman 09; Jegerlehner + Szafron 11 (ρ γ-mixing), also included in HLS-approach by Benayoun et al. 12. Lattice QCD: Various groups are working on it, precision at level of 5-10%, not yet competitive with phenomenological evaluations.

16 Muon g 2: current status Summary of SM contributions to a µ (based on various recent sources): Leptonic QED contributions: a QED µ = ( ± 0.036) Weak contributions: aµ weak = (153.2 {z } ± {z} 1.8 ) Hadronic contributions: - Vacuum Polarization:?? aµ HVP (e + e ) = ( ± 47.2 (100.3 ± 2.2)) 10 {z } {z}???? - Light-by-Light scattering: aµ HLbL = (116 ± 40) 10 {z } Total SM contribution:??? aµ SM = ( ± {z} 47 ± {z} 40 ± {z} 1.8 v.p. HLbL QED + EW [±62]) New experimental value (shifted due to new λ = µ µ/µ p): a exp µ = ( ± 63) a exp µ a SM µ = (294 ± 88) [3.3 σ] Sign of New Physics? Hadronic uncertainties need to be better controlled in order to fully profit from future g 2 experiments with δa µ =

17 Muon g 2: other recent evaluations HMNT (06) JN (09) Davier et al, τ (10) Davier et al, e + e (10) JS (11) HLMNT (10) HLMNT (11) experiment BNL BNL (new from shift in λ) 3.3 σ 2.4 σ 3.6 σ 3.3 σ 3.3 σ a µ Source: Hagiwara et al. 11. Note units of 10 10! Aoyama et al. 12: a exp µ a SM µ = (249 ± 87) [2.9 σ] Benayoun et al. 13: a exp µ a SM µ = [4.9 σ]

18 Part II: Hadronic light-by-light scattering in the muon g 2

19 Hadronic light-by-light scattering in the muon g 2 O(α 3 ) hadronic contribution to muon g 2: four-point function VVVV projected onto a µ (external soft photon k µ 0 0). k k = p p µ (p ) µ (p) Consider matrix element of light-quark electromagnetic current between muon states: j ρ(x) = 2 3 (ūγρu)(x) 1 3 ( dγ ρd)(x) 1 3 ( sγρs)(x) µ (p ) (ie)j ρ(0) µ (p) = ( ie)ū(p )Γ ρ(p, p)u(p) Z d 4 Z q 1 d 4 q 2 ( i) 3 i i = (2π) 4 (2π) 4 q1 2 q2 2 (q 1 + q 2 k) 2 (p q 1 ) 2 m 2 (p q 1 q 2 ) 2 m 2 ( ie) 3 ū(p )γ µ ( p q 1 + m)γ ν ( p q 1 q 2 + m)γ λ u(p) (ie) 4 Π µνλρ (q 1, q 2, k q 1 q 2 ) with k = p p and the fourth-rank light-quark hadronic tensor Z Z Z Π µνλρ (q 1, q 2, q 3 )= d 4 x 1 d 4 x 2 d 4 x 3 e i(q 1 x 1 +q 2 x 2 +q 3 x 3 ) Ω T{j µ(x 1 )j ν(x 2 )j λ (x 3 )j ρ(0)} Ω Momentum conservation: k = q 1 + q 2 + q 3.

20 Projection onto g 2, Properties of Π µνλρ (q 1, q 2, q 3 ) Flavor diagonal current j µ(x) is conserved and one has the Ward identities: {q µ 1 ; qν 2 ; qλ 3 ; kρ } Π µνλρ (q 1, q 2, q 3 ) = 0 Π µνλρ (q 1, q 2, k q 1 q 2 ) = k σ k ρ Π µνλσ(q 1, q 2, k q 1 q 2 ) Defining Γ ρ(p, p) = k σ Γ ρσ(p, p) one finally obtains (Aldins et al. 70): a µ = F 2 (0) = 1 48m tr (( p + m)[γρ, γ σ ]( p + m)γ ρσ(p, p)) i.e. one can formally take the limit k µ 0 inside the loop integrals. Problem reduces to calculation of two-point function with zero-momentum insertion. One also gets better UV convergence properties of individual Feynman diagrams (fermion-loop).

21 Projection onto g 2, Properties of Π µνλρ (q 1, q 2, q 3 ) Flavor diagonal current j µ(x) is conserved and one has the Ward identities: {q µ 1 ; qν 2 ; qλ 3 ; kρ } Π µνλρ (q 1, q 2, q 3 ) = 0 Π µνλρ (q 1, q 2, k q 1 q 2 ) = k σ k ρ Π µνλσ(q 1, q 2, k q 1 q 2 ) Defining Γ ρ(p, p) = k σ Γ ρσ(p, p) one finally obtains (Aldins et al. 70): a µ = F 2 (0) = 1 48m tr (( p + m)[γρ, γ σ ]( p + m)γ ρσ(p, p)) i.e. one can formally take the limit k µ 0 inside the loop integrals. Problem reduces to calculation of two-point function with zero-momentum insertion. One also gets better UV convergence properties of individual Feynman diagrams (fermion-loop). Properties of Π µνλρ (q 1, q 2, q 3 ) (Bijnens et al. 95): In general 138 Lorentz structures. But only 32 contribute to g 2. Using Ward identities, there are 43 gauge invariant structures. Bose symmetry relates some of them. All depend on q 2 1, q2 2, q2 3, q i q j, but before taking derivative and k µ 0, also on k 2, k q i. Compare with HVP: one function, one variable.

22 HLbL in muon g 2 QED: light-by-light scattering at higher orders in perturbation series via lepton-loop: γ γ e γ γ In muon g 2: Hadronic light-by-light scattering in muon g 2 from strong interactions (QCD): γ µ π + π 0, η, η e γ a had.lxl µ = = µ γ Coupling of photons to hadrons, e.g. π 0, via form factor: π 0 γ View before 2014: in contrast to HVP, no direct relation to experimental data size and even sign of contribution to a µ unknown! Approach: use some hadronic model at low energies with exchanges and loops of resonances and some form of (dressed) quark-loop at high energies. Problem: VVVV depends on several invariant momenta distinction between low and high energies is not as easy as for two-point function VV (HVP). Note: one can (in principle) always perform Wick rotation to Euclidean momenta, where effects of resonances are smoothed out, but there are mixed regions where Q1 2 is small and Q2 2 large and vice versa.

23 Hadronic light-by-light scattering Data on γγ γγ In any case, it is a good idea to look at actual data. For instance, obtained by Crystal Ball detector 88 via e + e e + e γ γ e + e π 0 : Invariant γγ mass spectrum: e π 0 e + Feynman diagram from Colangelo et al., arxiv: Three spikes from light pseudoscalars in reaction: γγ π 0, η, η γγ for (almost) real photons.

24 Hadronic light-by-light scattering Data on γγ γγ In any case, it is a good idea to look at actual data. For instance, obtained by Crystal Ball detector 88 via e + e e + e γ γ e + e π 0 : Invariant γγ mass spectrum: e π 0 e + Feynman diagram from Colangelo et al., arxiv: Three spikes from light pseudoscalars in reaction: γγ π 0, η, η γγ for (almost) real photons. New development in 2014: use of dispersion relations for a data driven approach to HLbL in muon g 2 from (still to be measured!) scattering of two off-shell photons: γ γ π 0, η, η γ γ ππ (Colangelo et al.; Pauk, Vanderhaeghen).

25 Current approach to HLbL scattering in g 2 Classification of de Rafael 94 Chiral counting p 2 (from Chiral Perturbation Theory (ChPT)) and large-n C counting as guideline to classify contributions (all higher orders in p 2 and N C contribute): µ (p ) k k = p p µ (p) = ρ π + π 0,, η, η Exchange of other resonances + (f 0, a 1, f 2...) + ρ Q Chiral counting: p 4 p 6 p 8 p 8 N C -counting: 1 N C N C N C pion-loop pseudoscalar exchanges quark-loop (dressed) (dressed) Relevant scales in HLbL ( VVVV with off-shell photons!): 0 2 GeV, i.e. much larger than m µ!

26 Current approach to HLbL scattering in g 2 Classification of de Rafael 94 Chiral counting p 2 (from Chiral Perturbation Theory (ChPT)) and large-n C counting as guideline to classify contributions (all higher orders in p 2 and N C contribute): µ (p ) k k = p p µ (p) = ρ π + π 0,, η, η Exchange of other resonances + (f 0, a 1, f 2...) + ρ Q Chiral counting: p 4 p 6 p 8 p 8 N C -counting: 1 N C N C N C pion-loop pseudoscalar exchanges quark-loop (dressed) (dressed) Relevant scales in HLbL ( VVVV with off-shell photons!): 0 2 GeV, i.e. much larger than m µ! Constrain models using experimental data (processes of hadrons with photons: decays, form factors, scattering) and theory (ChPT at low energies; short-distance constraints from pqcd / OPE at high momenta). Issue: on-shell versus off-shell form factors. For instance pion-pole with on-shell pion form factors versus pion-exchange with off-shell pion form factors (in both cases with one or two off-shell photons). Pseudoscalars: numerically dominant contribution (according to most models!). Warrant special attention.

27 Pion-pole versus pion-exchange in a HLbL;π0 µ To uniquely identify contribution of exchanged neutral pion π 0 in Green s function VVVV, we need to pick out pion-pole: q 1 q π crossed diagrams q 2 lim (q 1 +q 2 ) 2 m 2 π ((q 1 + q 2 ) 2 m 2 π) VVVV Residue of pole: on-shell vertex function 0 VV π on-shell form factor F π 0 γ γ (q2 1, q2 2 ) q 3

28 Pion-pole versus pion-exchange in a HLbL;π0 µ To uniquely identify contribution of exchanged neutral pion π 0 in Green s function VVVV, we need to pick out pion-pole: q 1 q π crossed diagrams q 2 lim ((q 1 + q 2 ) 2 mπ) VVVV 2 (q 1 +q 2 ) 2 mπ 2 Residue of pole: on-shell vertex function 0 VV π on-shell form factor F π 0 γ γ (q2 1, q2 2 ) But in contribution to muon g 2, we evaluate Feynman diagrams, integrating over photon momenta with exchanged off-shell pions. For all the pseudoscalars: q 3 π 0, η, η,... Shaded blobs represent off-shell form factor F PS γ γ ((q 1 + q 2 ) 2, q 2 1, q2 2 ) where PS = π 0, η, η, π 0,... Off-shell form factors are either inserted by hand starting from constant, pointlike Wess-Zumino-Witten (WZW) form factor or using e.g. some resonance Lagrangian. Contribution with off-shell form factors is model dependent. Within a dispersive framework the pole contribution can be uniquely defined from imaginary part of Feynman diagram (similarly for multi-particle intermediate states like ππ), but this represents only part of the contribution.

29 HLbL in muon g 2: main results

30 HLbL scattering: Summary of selected results µ (p ) µ (p) k k = p p = ρ π + π 0,, η, η Exchange of other resonances + (f 0, a 1, f 2...) Q + ρ Chiral counting: p 4 p 6 p 8 p 8 N C -counting: 1 N C N C N C Contribution to a µ : BPP: +83 (32) HKS: +90 (15) KN: +80 (40) MV: +136 (25) 2007: +110 (40) PdRV:+105 (26) N,JN: +116 (40) -19 (13) -5 (8) 0 (10) ud.: (19) -19 (13) +85 (13) +83 (6) +83 (12) +114 (10) +114 (13) +99 (16) ud.: + ud. = undressed, i.e. point vertices without form factors -4 (3) [f 0, a 1] +1.7 (1.7) [a 1] +22 (5) [a 1] +8 (12) [f 0, a 1] +15 (7) [f 0, a 1] +21 (3) +10 (11) [c-quark] +21 (3) ud.: +60 BPP = Bijnens, Pallante, Prades 96, 02; HKS = Hayakawa, Kinoshita, Sanda 96, 98, 02; KN = Knecht, Nyffeler 02; MV = Melnikov, Vainshtein 04; 2007 = Bijnens, Prades; Miller, de Rafael, Roberts; PdRV = Prades, de Rafael, Vainshtein 09 (compilation); N,JN = Nyffeler 09; Jegerlehner, Nyffeler 09 (compilation) Recall (in units of ): δa µ(hvp) 45; δa µ(exp [BNL]) = 63; δa µ(future exp) = 16

31 HLbL scattering: Summary of selected results (continued) Some results for the various contributions to a HLbL µ : Contribution BPP HKS, HK KN MV BP, MdRR PdRV N, JN π 0, η, η 85± ±6.4 83±12 114±10 114±13 99 ± 16 axial vectors 2.5± ±1.7 22±5 15±10 22±5 scalars 6.8±2.0 7±7 7±2 π, K loops 19±13 4.5±8.1 19±19 19±13 π,k loops +subl. N C 0±10 quark loops 21±3 9.7± (c-quark) 21±3 Total 83± ± ±40 136±25 110± ± ± 39 BPP = Bijnens, Pallante, Prades 95, 96, 02; HKS = Hayakawa, Kinoshita, Sanda 95, 96; HK = Hayakawa, Kinoshita 98, 02; KN = Knecht, Nyffeler 02; MV = Melnikov, Vainshtein 04; BP = Bijnens, Prades 07; MdRR = Miller, de Rafael, Roberts 07; PdRV = Prades, de Rafael, Vainshtein 09; N = Nyffeler 09, JN = Jegerlehner, Nyffeler 09 Pseudoscalar-exchanges dominate numerically. Other contributions not negligible. Cancellation between π, K-loops and quark loops! PdRV: Analyzed results obtained by different groups with various models and suggested new estimates for some contributions (shifted central values, enlarged errors). Do not consider dressed light quark loops as separate contribution! Assume it is already taken into account by using short-distance constraint of MV 04 on pseudoscalar-pole contribution. Added all errors in quadrature! N, JN: New evaluation of pseudoscalar exchange contribution imposing new short-distance constraint on off-shell form factors. Took over most values from BPP, except axial vectors from MV. Added all errors linearly.

32 Comments History: several changes in size and even sign due to errors since first evaluation by Calmet et al : last sign change in dominant pseudoscalar contribution (KN 02). More recently, size of dressed quark-loop and dressed pion-loop have been questioned (Fischer et al. 11, 13; Engel et al. 12, 13). All estimates are currently based on models! (ENJL, VMD, large-n C QCD matched to OPE (LMD, LMD+V), HLS, (constituent, chiral) quark models, AdS/QCD, Padé approximants, DSE with truncations... ). In some cases it is not obvious whether actually the same contributions are calculated or not, e.g. pion-pole versus pion-exchange. Currently most used estimates: aµ HLbL = (105 ± 26) (Prades, de Rafael, Vainshtein 09) ( Glasgow consensus ) a HLbL µ = (116 ± 40) (Nyffeler 09; Jegerlehner, Nyffeler 09) Compilations based on work by different groups using various models. Error estimates are mostly guesses! Data driven approach to HLbL with a more controlled error estimate is needed, e.g. using dispersion relations, similarly to HVP! Future: maybe lattice QCD will provide one day a reliable estimate for HLbL in muon g 2. Preliminary attempts by Blum et al. 05,..., 14.

33 One-particle intermediate states: resonance exchanges / poles Note Within phenomenological approach with models, one particle exchanges / poles come from pseudoscalars, scalars, axial vectors and tensor mesons. Within dispersion relation approach only stable physical intermediate states are considered: one-pion, two-pion, three-pion, photons, muons...

34 Pion-pole contribution Pion-pole contribution determined by measurable pion transition form factor F π 0 γ γ (q2 1, q2 2 ) (on-shell pion, one or two off-shell photons). Analogously for η, η -pole contributions. Knecht, Nyffeler 02: Z d a HLbL;π0 µ = e 6 4 q 1 d 4 q 2 1 (2π) 4 (2π) 4 q1 2q2 2 (q1 + q2)2 [(p + q 1) 2 mµ 2 ][(p q2)2 mµ 2 ] " Fπ 0 γ γ (q 2 1, (q1 + q2)2 ) F π 0 γ γ (q 2 2, 0) q2 2 T 1(q 1, q 2; p) m2 π + F π 0 γ γ (q2 1, q2 2 ) F π 0 γ γ ((q1 + q2)2, 0) (q 1 + q 2) 2 m 2 π T 2(q 1, q 2; p) # T 1(q 1, q 2; p) = 16 3 (p q1) (p q2) (q1 q2) 16 3 (p q2)2 q (p q1) (q1 q2) q (p q 2) q 2 1 q (p q2) (q1 q2) m2 µ q2 1 q m2 µ (q1 q2)2 T 2(q 1, q 2; p) = 16 3 (p q1) (p q2) (q1 q2) 16 3 (p q1)2 q (p q1) (q1 q2) q (p q1) q2 1 q m2 µ q2 1 q m2 µ (q1 q2)2 where p 2 = m 2 µ and the external photon has now zero four-momentum (soft photon). Currently, only single-virtual TFF F π 0 γ γ (Q2, 0) has been measured by CELLO, CLEO, BABAR, Belle. Analysis ongoing at BES, measurement planned at KLOE-2.

35 Relevant momentum regions in a HLbL;π0 µ In Knecht, Nyffeler 02, a 2-dimensional integral representation for the pion-pole contribution was derived for a certain class of form factors (VMD-like). Schematically: Z Z X aµ HLbL;π0 = dq 1 dq 2 w i (Q 1, Q 2 ) f i (Q 1, Q 2 ) 0 0 i with universal weight functions w i. Dependence on form factors resides in the f i. Plot of weight functions w i from Knecht, Nyffeler 02: Q 2 [GeV] Q [GeV] 2 0 w f1 (Q 1,Q 2 ) w g2 (M π,q 1,Q 2 ) 1 Q 1 [GeV] Q 1 [GeV] Q [GeV] 2 1 Q 2 [GeV] w g1 (M V,Q 1,Q 2 ) w g2 (M V,Q 1,Q 2 ) 1 Q [GeV] Q 1 [GeV] 2 Relevant momentum regions around GeV. As long as form factors in different models lead to damping, expect comparable results for aµ HLbL;π0, at level of 20%. Jegerlehner, Nyffeler 09 derived 3-dimensional integral representation for general form factors (hyperspherical approach). Integration over Q1 2, Q2 2, cos θ, where Q 1 Q 2 = Q 1 Q 2 cos θ. Idea recently taken up by Dorokhov et al. 12 (for scalars) and Bijnens, Zahiri Abyaneh 12, 13 (for all contributions, work in progress).

36 Impact of form factor measurements: example KLOE-2 ) 2 F(Q On the possibility to measure the π 0 γγ decay width and the γ γ π 0 transition form factor with the KLOE-2 experiment D. Babusci, H. Czyż, F. Gonnella, S. Ivashyn, M. Mascolo, R. Messi, D. Moricciani, A. Nyffeler, G. Venanzoni and the KLOE-2 Collaboration γ*γ π Q 2 [GeV] Simulation of KLOE-2 measurement of TFF F (Q 2 ) (red triangles). MC program EKHARA 2.0 (Czyż, Ivashyn 11) and detailed detector simulation. Solid line: F (0) given by chiral anomaly (WZW). Dashed line: form factor according to on-shell LMD+V model (Knecht, Nyffeler 01). CELLO (black crosses) and CLEO (blue stars) data at higher Q 2. Within 1 year of data taking, collecting 5 fb 1, KLOE-2 will be able to measure: Γ π 0 γγ to 1% statistical precision. γ γ π 0 transition form factor F (Q 2 ) in the region of very low, space-like momenta 0.01 GeV 2 Q GeV 2 with a statistical precision of less than 6% in each bin. KLOE-2 can (almost) directly measure slope of form factor at origin (note: logarithmic scale in Q 2 in plot!).

37 Impact of form factor measurements: example KLOE-2 (continued) Error in aµ HLbL;π0 related to model parameters determined by Γ π 0 γγ (normalization of FF; not taken into account in most papers) and F (Q 2 ) will be reduced as follows: δaµ HLbL;π (with current data for F (Q 2 ) + Γ PDG π 0 γγ ) δa HLbL;π0 µ (+ Γ PrimEx π 0 γγ ) δa HLbL;π0 µ ( ) (+ KLOE-2 data) Note that error does not account for other potential uncertainties in a HLbL;π0 µ, e.g. related to choice of model, 2nd off-shell photon, off-shellness of pion. Simple models with few parameters, like VMD (two parameters: F π, M V ), which are completely determined by the data on Γ π 0 γγ and F (Q2 ), can lead to very small errors in a HLbL;π0 µ. For illustration: a HLbL;π0 µ;vmd = (57.3 ± 1.1) a HLbL;π0 µ;lmd+v = (72 ± 12) (off-shell LMD+V FF, including all errors) But this might be misleading! VMD and LMD+V give equally good fits to transition form factor F (Q 2 ), but differ in doubly-off shell transition form factor F π 0 γ γ (q2 1, q2 2 ). Results for ahlbl;π0 µ differ by about 20%! Reason: VMD form factor has wrong high-energy behavior too strong damping in a HLbL;π0 µ;vmd : F VMD π 0 γ γ (q 2, q 2 ) 1/q 4, for large q 2, i.e. falls off too fast compared to OPE prediction F OPE π 0 γ γ (q 2, q 2 ) 1/q 2 which is fulfilled by LMD+V. Dispersive approach to not rely (or less) on models.

38 Pseudoscalar pole / exchange in large-n C QCD Model for F P ( ) γ γ a µ(π 0 ) a µ(π 0, η, η ) VMD LMD (on-shell) [KN] 73 LMD+V (on-shell, h 2 = 0) [KN] 58( 10 ) 83( 12 ) LMD+V (on-shell, h 2 = 10 GeV 2 ) [KN] 63( 10 ) 88( 12 ) LMD+V (on-shell, constant 2nd FF) [MV] 77( 7 ) 114( 10 ) LMD+V (off-shell) [N] 72( 12 ) 99( 16 ) LMD+P (off-shell) [KaNo] 65.8(1.2) LMD+P (off-shell) [RGL] 66.5(1.9) 104.3(5.2) LMD+P (on-shell) [RGL] 57.5(0.5) 82.7(2.8) KN = Knecht, Nyffeler 02 MV = Melnikov, Vainshtein 04 N = Nyffeler 09 KaNo = Kampf, Novotny 11 (RχT) RGL = Roig, Guevara, Lopez Castro 14 (RχT) Note: KN, MV, N use VMD FF for η, η

39 Other recent partial evaluations (mostly pseudoscalars) Nonlocal chiral quark model (off-shell) [Dorokhov et al.] 2008: a LbyL;π0 µ = 65(2) : a LbyL;π0 µ = 50.1(3.7) 10 11, a LbyL;PS µ = 58.5(8.7) : a LbyL;π0 +σ µ = 54.0(3.3) 10 11, a LbyL;a 0+f 0 µ a LbyL;PS+S µ = 62.5(8.3) Strong damping for off-shell form factors. Positive and small contribution from scalar σ(600), differs from other estimates (BPP 96, 02; Blokland, Czarnecki, Melnikov 02). Holographic (AdS/QCD) model 1 (off-shell?) [Hong, Kim 09] aµ LbyL;π0 = , aµ LbyL;PS = Holographic (AdS/QCD) model 2 (off-shell) [Cappiello, Cata, D Ambrosio 10] a LbyL;π0 µ = 65.4(2.5) Used AdS/QCD to fix parameters in ansatz by D Ambrosio et al. 98. Padé approximants (on-shell) a LbyL;π0 µ = 54(5) [Masjuan 12 (using on-shell LMD+V FF)] a LbyL;π0 µ = 64.9(5.6) 10 11, a LbyL;PS µ = 89(7) [Escribano, Masjuan, Sanchez-Puertas 13] Fix parameters in Padé approximants from data on transition form factors.

40 More on single-resonance poles and exchanges Pauk, Vanderhaeghen 14: a µ(f 0 (980), f 0, a 0) = [( 0.9 ± 0.2) to ( 3.1 ± 0.8)] a µ(f 1, f 1 ) = (6.4 ± 2.0) a µ(f 2, f 2 2, a 2 ) = (1.1 ± 0.1) Meson pole approximation. Using monopole form factor (VMD) for scalars and dipole form factors (strong damping!) for axial vectors and tensors: F mon Mγ γ (q2 1, q2 2 ) F Mγ γ (0, 0) F dip Mγ γ (q2 1, q2 2 ) F Mγ γ (0, 0) = = 1 1 (1 q 1 2/Λ2 mon ) (1 q 2 2/Λ2 mon ) 1 (1 q 2 1 /Λ2 dip )2 1 (1 q 2 2 /Λ2 dip )2 First calculation of tensor meson contribution to HLbL. Forward sum rules for γγ scattering (Pascalutsa, Vanderhaeghen, 10; Pascalutsa, Pauk, Vanderhaeghen 12) used to constrain model. They link transition form factors of pseudoscalars, axial vectors and tensor mesons Λ dip = 1.5 GeV. Jegerlehner (Talks at g 2 Workshops in Mainz, April 2014): a µ(f 0 (980), f 0, a 0) = ( ( ) ± 1.20) a µ(f 1, f 1, a 1) = ( ) ± 2.71) Meson exchange contributions. Models inspired from Melnikov, Vainshtein 04, but correctly implement Landau-Yang theorem. Both calculations give contribution for axial vectors, which is substantially smaller than Melnikov, Vainshtein 04: a µ(f 1, f 1, a 1) = (22 ± 5)

41 Two-particle intermediate states (pion-loop)

42 Dressed pion-loop 1. ENJL/VMD versus HLS Model a π loop µ scalar QED (no FF) -45 HLS -4.5 ENJL -19 full VMD -15 1e-10 8e-11 6e-11 4e-11 LLQ a µ 2e-11 Origin: different behavior of integrands in contribution to g 2 Strong damping if form factors are introduced, very model dependent: compare ENJL (BPP 96) versus HLS (HKS 96). See also discussion in Melnikov, Vainshtein 04. (Zahiri Abyaneh 12; Bijnens, Zahiri Abyaneh 12; Talks by Bijnens at MesonNet 2013, Prague; g 2 Workshop, Mainz 2014) 0-2e-11-4e-11 p 2 α p 4 p 3 β p 1 ν p 5 q ρ p p π loop Q One can do 5 of the 8 integrations in the 2-loop integral for g 2 analytically, using the hyperspherical approach / Gegenbauer polynomials (Jegerlehner, Nyffeler 09; taken up in VMD HLS a=2 Bijnens, Zahiri Abyaneh 12): a µ X Z = dl P1 dl P2 a µ XLL Z = dl P1 dl P2 dl Q a µ XLLQ, with l P = ln(p/gev) Contribution of type X at given scale P 1, P 2, Q is directly proportional to volume under surface when a µ XLL and aµ XLLQ are plotted versus the energies on a logarithmic scale P 1 = P 2 Momentum distribution of full VMD and HLS pion-loop contribution for P 1 = P 2. HLS: Integrand changes from positive to negative at high momenta. Leads to cancellation and therefore smaller absolute value. Usual HLS model (a = 2) known to not fullfill certain QCD short-distance constraints.

43 Dressed pion-loop (continued) 2. Role of pion polarizability and a 1 resonance Engel, Patel, Ramsey-Musolf 12: ChPT analysis of LbyL up to order p 6 in limit p 1, p 2, q m π. Identified potentially large contributions from pion polarizability (L 9 + L 10 in ChPT) which are not fully reproduced in ENJL / HLS models used by BPP 96 and HKS 96. Pure ChPT approach is not predictive. Loops not finite, would need new a µ counterterm (Knecht et al. 02). Engel, Ph.D. Thesis 13; Engel, Ramsey-Musolf 13: tried to include a 1 resonance explicitly in EFT. Problem: contribution to g 2 in general not finite (loops with resonances) Form factor approach with a 1 that reproduces pion polarizability at low energies, has correct QCD scaling at high energies and generates a finite result in a µ: L I = e2 4 Fµνπ+ 1 " L II = e2 2MA 2 π + π! D 2 + MA 2 F µν π + h.c. +! # M 2 2 V 2 + MV 2 F µν + a π loop µ : Model (a) (b) I II Second and third columns in Table correspond to different values for the polarizability LECs, (α r 9 + αr 10 ): (a) (1.32 ± 1.4) 10 3 (from radiative pion decay π + e + ν eγ ) and (b) (3.1 ± 0.9) 10 3 (from radiative pion photoproduction γp γ π + n). Potentially large results (absolute value): a π loop µ (11 71) Variation of ! Uncertainty underestimated in earlier calculations?

44 Dressed pion-loop (continued) Issue taken up in Zahiri Abyaneh 12; Bijnens, Zahiri Abyaneh 12; Bijnens, Relefors (to be published); Talk by Bijnens at g 2 Workshop, Mainz Tried various ways to include a 1, but again no finite result for g 2 achieved. With a cutoff of 1 GeV: a π loop µ = ( 20 ± 5) (preliminary) Very close to old result a π loop µ = ( 19 ± 13) in BPP 96, 02. Maybe only model-independent approach with dispersion relations can give a reliable prediction for pion-loop. But to include the effects of the axial vector meson a 1 and the matching with QCD short-distances, the region 1 2 GeV should be covered as well.

45 Dressed quark-loop

46 Dressed quark-loop Dyson-Schwinger equation (DSE) approach [Fischer, Goecke, Williams 11, 13] Claim: no double-counting between quark-loop and pseudoscalar exchanges (or exchanges of other resonances) HLbL in Effective Field Theory (hadronic) picture: q + + +/ + Quarks here may have different interpretation than below! HLbL using functional methods (all propagators and vertices fully dressed):... q Expansion of quark-loop in terms of planar diagrams (rainbow-ladder approximation): q = + + Pole representation of ladder-exchange contribution:... P 2 M 2 PS Truncate DSE using well tested model for dressed quark-gluon vertex (Maris, Tandy 99). Large contribution from quark-loop (even after recent correction), in contrast to all other approaches, where coupling of (constituent) quarks to photons is dressed by form factors (ρ γ-mixing, VMD).

47 Dressed quark-loop (continued) Dyson-Schwinger equation approach [Fischer, Goecke, Williams 11, 13] a HLbL;π0 µ = 57.5(6.9) (off-shell) a HLbL;PS µ = 81(2) a HLbL;quark loop µ = 107(2) a HLbL µ = 188(4) Error for PS, quark-loop and total only from numerics. Quark-loop: some parts are missing. Not yet all contributions to HLbL calculated. Systematic error? Note: numerical error in quark-loop in earlier paper (GFW PRD83 11): a HLbL;quark loop µ = 136(59) a HLbL µ = 217(91) 10 11

48 Dressed quark-loop (continued) Constituent quark loop [Boughezal, Melnikov 11] a HLbL µ = ( ) Consider ratio of HVP and HLbL with pqcd corrections. Paper was reaction to earlier results using DSE yielding large values for quark-loop and total. Constituent Chiral Quark Model [Greynat, de Rafael 12] a HLbL;CQloop µ = 82(6) a HLbL;π0 µ = 68(3) (off-shell) a HLbL µ = 150(3) Error only reflects variation of constituent quark mass M Q = 240 ± 10 MeV, fixed to reproduce HVP in g 2. Determinations from other quantities give larger value for M Q 300 MeV and thus smaller value for quark-loop. 20%-30% systematic error estimated. Not yet all contributions calculated. Padé approximants [Masjuan, Vanderhaeghen 12] a HLbL µ = (76(4) 125(7)) Quark-loop with running mass M(Q) ( ) MeV, where average momentum Q ( ) MeV is fixed from relevant momenta in 2-dim. integral representation for pion-pole in Knecht, Nyffeler 02. µ q3 q2 q1 µ + P π

49 Summary of recent developments Recent evaluations for pseudoscalars: a HLbL;π0 µ (50 69) a HLbL;PS µ (59 107) Most evaluations agree at level of 15%, but some are quite different. New estimates for axial vectors (Pauk, Vanderhaeghen 14; Jegerlehner 14): a HLbL;axial µ (6 8) Substantially smaller than in MV 04! First estimate for tensor mesons (Pauk, Vanderhaeghen 14): a HLbL;tensor µ = (1.1 ± 0.1) Open problem: Dressed pion-loop Potentially important effect from pion polarizability and a 1 resonance (Engel, Patel, Ramsey-Musolf 12; Engel 13; Engel, Ramsey-Musolf 13): a HLbL;π loop µ = (11 71) Maybe large negative contribution, in contrast to BPP 96, HKS 96. Open problem: Dressed quark-loop Dyson-Schwinger equation approach (Fischer, Goecke, Williams 11, 13): a HLbL;quark loop µ = (still incomplete!) Large contribution, no damping seen, in contrast to BPP 96, HKS 96.

50 Data driven approach to HLbL using dispersion relations (DR)

51 Strategy: Split contributions to HLbL into two parts I: Data-driven evaluation using DR (hopefully numerically dominant): (1) π 0, η, η poles (2) ππ intermediate state II: Model dependent evaluation (hopefully numerically subdominant): (1) Axial vectors (3π-intermediate state),... (2) Quark-loop, matching with pqcd Error goals: Part I: 10% precision (data driven), Part II: 30% precision. To achieve overall error of about 20% (δa HLbL µ = ).

52 Strategy: Split contributions to HLbL into two parts I: Data-driven evaluation using DR (hopefully numerically dominant): (1) π 0, η, η poles (2) ππ intermediate state II: Model dependent evaluation (hopefully numerically subdominant): (1) Axial vectors (3π-intermediate state),... (2) Quark-loop, matching with pqcd Error goals: Part I: 10% precision (data driven), Part II: 30% precision. To achieve overall error of about 20% (δa HLbL µ = ). Colangelo et al., arxiv: , arxiv: : Classify intermediate states in four-point function. Then project onto g 2. Π µνλσ = Π π0 µνλσ + ΠFsQED µνλσ + Πππ µνλσ + Π π0 µνλσ = pion pole (similarly for η, η ). Contribution to g 2 discussed earlier. Π FsQED µνλσ = scalar QED with vertices dressed by pion vector form factor F π V Π ππ µνλσ = remaining ππ contribution

53 Photon-photon processes in e + e collisions Feynman diagrams from Colangelo et al., arxiv: Space-like kinematics: e π 0 e + e e + π π By tagging the outgoing leptons (single-tag, double-tag), one can infer the virtual (space-like) momenta Q 2 i = q 2 i of the photons. Left: process allows to measure pion-photon-photon transition form factor (TFF) F π 0 γ γ (Q2 1, Q2 2 ). Time-like kinematics: e π 0 e + e e + π π

54 ππ intermediate state Colangelo et al., arxiv: , arxiv: Extracting the pion vector form factor Fπ V from data, the contribution ΠFsQED µνλσ is known to sufficient accuracy. The remaining ππ contribution from Π ππ µνλσ is then given by Z d aµ ππ 4 Z = q 1 d 4 P q 2 e6 i I i `s, q 2 1, q2 2 T ππ i `q1, q 2 ; p (2π) 4 (2π) 4 q1 2q2 2 sz, 1Z 2 with known kinematical functions Ti ππ (q 1, q 2 ; p), while information on scattering amplitude on the cut is given by the dispersive integrals I i (s, q1 2, q2 2 ).

55 ππ intermediate state Colangelo et al., arxiv: , arxiv: Extracting the pion vector form factor Fπ V from data, the contribution ΠFsQED µνλσ is known to sufficient accuracy. The remaining ππ contribution from Π ππ µνλσ is then given by Z d aµ ππ 4 Z = q 1 d 4 P q 2 e6 i I i `s, q 2 1, q2 2 T ππ i `q1, q 2 ; p (2π) 4 (2π) 4 q1 2q2 2 sz, 1Z 2 with known kinematical functions Ti ππ (q 1, q 2 ; p), while information on scattering amplitude on the cut is given by the dispersive integrals I i (s, q1 2, q2 2 ). For instance, for first S-wave: Z 2 I 1`s, q 1, 1 ds» 1 q2 2 = π s s s s s q 2 1 «q2 2 λ`s, q 2 4Mπ 2 1, Im h 0 q2 ++,++ 2 2ξ 1ξ 2 + λ`s, q1 2, q2 2 λ(x, y, z) = x 2 + y 2 + z 2 2(xy + xz + yz): Källén function ξ i : normalization of longitudinal polarization vectors of off-shell photons Im h 0 `s 00,++ ; q 2 1, q2 2 ; s, 0 `s ; q 2 1, q2 2 ; s, 0 hλ J 1 λ 2,λ 3 λ 4 (s; q1 2, q2 2 ; q2 3, q2 4 ): partial-wave helicity amplitudes with angular momentum J for process γ `q 1, λ 1 γ `q 2, λ 2 γ `q 3, λ 3 γ `q 4, λ 4. Partial-wave unitarity relates imaginary parts in integrals I i to helicity partial waves h J,λ1 λ 2 (s; q1 2, q2 2 ) for γ γ ππ, which have to be determined from experiment: p Im hλ J 1 λ 2,λ 3 λ 4 `s; q 2 1 4M 1, q2 2 ; q2 3, 2 4 q2 = π /s h J,λ1 λ 16π 2 `s; q 2 1, q2 hj,λ3 2 λ 4 `s; q 2 3, q4 2

56 Another approach to HLbL using DR s Pauk, Vanderhaeghen, arxiv: , arxiv: Write DR directly for form factor F 2 (k 2 ): F 2 (0) = 1 2πi Z 0 dk 2 k 2 Disc k 2 F 2(k 2 ) Then study contributions from different intermediate states from the cuts in the unitarity diagrams related to measurable physical processes like γ γ X and γ γx :

57 Another approach to HLbL using DR s Pauk, Vanderhaeghen, arxiv: , arxiv: Write DR directly for form factor F 2 (k 2 ): F 2 (0) = 1 2πi Z 0 dk 2 k 2 Disc k 2 F 2(k 2 ) Then study contributions from different intermediate states from the cuts in the unitarity diagrams related to measurable physical processes like γ γ X and γ γx : Pseudoscalar pole contribution Considering the pseudoscalar intermediate state and VMD form factor (ρ γ mixing) it was shown that from dispersion relation one obtains precisely the result given by direct evaluation of two-loop integral in Knecht, Nyffeler 02 for the pseudoscalar pole. Dashed: two-particle cut Dotted: three-particle cut Double-dashed: pseudoscalar pole Double-solid: vector meson pole

58 NLO Golangelo, Hoferichter, Nyffeler, Passera, Stoffer 14 Recently, a surprisingly large NNLO HVP contribution was obtained by Kurz et al. 14: HVP, LO aµ = ( ± 47.2) HVP, NLO aµ = ( ± 2.2) HVP, NNLO aµ = (12.4 ± 0.1) Enhancement because of large ln(m µ/m e), prefactors π 2 in QED LbL. Could there be a similarly large effect in NLO? We calculated the potentially large contribution from an additional electron loop (using simple VMD model for pion-pole to model full HLbL) a π0 -pole, NLO µ = % effect of a π0 -pole µ = , very close to renormalization group arguments 3 α π 2 mµ log 2.5%. 3 m e (a) 3a (b) 3b (c) 3b (d) 3c (e) 3c (f) 3c (g) 3b,lbl (h) 3d Figure 1: Sample NNLO Feynman diagrams contributing to a had µ. Estimating the not yet calculated diagrams with HLbL with additional radiative corrections to the muon line or internal HLbL by comparing with HLbL insertions with muon loop in QED, which are suppressed by factor 4, we obtain the estimate: HLbL, NLO aµ = (3 ± 2) 10 11, Negligible with current precision goal for HLbL!

59 Conclusions Over many decades, the (anomalous) magnetic moments of electron and muon have played a crucial role in atomic and elementary particle physics. a e: Test of QED, precise determination of fine-structure constant α. a µ: Test of Standard Model, potential window to New Physics: a exp µ a SM µ = (294 ± 88) [3.3 σ] Two new planned g 2 experiments at Fermilab and JPARC with goal of δaµ exp = (factor 4 improvement) Theory needs to match this precision! Hadronic vacuum polarization Ongoing and planned experiments on σ(e + e hadrons) with a goal of δaµ HVP = (20 25) (factor 2 improvement) Hadronic light-by-light scattering a HLbL µ = (105 ± 26) (Prades, de Rafael, Vainshtein 09) a HLbL µ = (116 ± 40) (Nyffeler 09; Jegerlehner, Nyffeler 09) Need a much better understanding of the complicated hadronic dynamics to get reliable error estimate of ± Better theoretical models needed; more constraints from theory (ChPT, pqcd, OPE); close collaboration of theory and experiment to measure the relevant decays, form factors and cross-sections of various hadrons with photons. - Promising new data driven approach using dispersion relations for π 0, η, η and ππ. Still needed: data for scattering of off-shell photons. - Future: Lattice QCD.

60 In 1959, six physicists joined forces to try to measure the muon s magnetic moment using CERN s first accelerator, the Synchrocyclotron. In 1961, the team published the first direct measurement of the muon s anomalous magnetic moment to a precision of 2% with respect to the theoretical value. By 1962, this precision had been whittled down to just 0.4%. This was a great success since it validated the theory of quantum electrodynamics. As predicted, the muon turned out to be a heavy electron. The second g-2 experiment started in 1966 under the leadership of Francis Farley and it achieved a precision 25 times higher than the previous one. This allowed phenomena predicted by the theory of quantum electrodynamics to be observed with a much greater sensitivity vacuum polarisation for instance, which is the momentary appearance of virtual electron and antielectron pairs with very short lifetimes. The experiment also revealed a quantitative discrepancy with the theory and thus prompted theorists to re-calculate their predictions. A third experiment, with a new technical approach, was launched in 1969, under the leadership of Emilio Picasso. The final results were published in 1979 and confirmed the theory to a precision of %. They also allowed observation of a phenomenon contributing to the magnetic moment, namely the presence of virtual hadrons. After 1984, the United States took up the mantle of investigating the muon s anomalous magnetic moment, applying the finishing touches to an edifice built at CERN. And finally: g-2 measuring the muon In the 1950s the muon was still a complete enigma. Physicists could not yet say with certainty whether it was simply a much heavier electron or whether it belonged to another species of particle. g-2 was set up to test quantum electrodynamics, which g-2 is not an experiment: it is a way of life. John Adams predicts, among other things, an anomalously high value for the muon s magnetic moment g, hence the name of the experiment. The first g-2 experiment at the SC, sitting on the experiment s 6 m long magnet. From right to left : A. Zichichi, Th. Muller, G. Charpak, J.C. Sens. Standing : F. Farley. The sixth person was R. Garwin. The g-2 muon storage ring in «The science I have experienced has been all about imagining and creating pioneering devices and observing entirely new phenomena, some of which have possibly never even been predicted by theory. That s what invention is all about and it s something quite extraordinary. CERN was marvellous for two reasons: it gave young people like me the opportunity to forge ahead in a new field and the chance to develop in an international environment.» Francis Farley, Infinitely CERN, 2004 Source: CERN g 2 is not an experiment: it is a way of life. John Adams (Head of the Proton Synchrotron at CERN ( ) and Director General of CERN ( )) This statement also applies to many theorists working on the g 2!

61 Backup

62 Form factor F π 0 γ γ (q2 1, q2 2 ) and transition form factor F (Q2 ) Form factor F π 0 γ γ (q2 1, q2) 2 between an on-shell pion and two off-shell photons: Z i d 4 x e iq1 x 0 T {j µ(x)j ν(0)} π 0 (q 1 + q 2) = ε µναβ q1 α q β 2 F π 0 γ γ (q2 1, q2) 2 0 j µ(x) = (ψ ˆQγ µψ)(x), u d A, ˆQ = diag(2, 1, 1)/3 s (light quark part of electromagnetic current) Bose symmetry: F π 0 γ γ (q2 1, q 2 2) = F π 0 γ γ (q2 2, q 2 1) Form factor for real photons is related to π 0 γγ decay width: 1 F 2 π 0 γ γ (q2 1 = 0, q2 2 4 = 0) = πα 2 mπ 3 Often normalization with chiral anomaly is used: Pion-photon transition form factor: 1 F π 0 γ γ (0, 0) = 4π 2 F π Γ π 0 γγ F (Q 2 ) F π 0 γ γ ( Q2, q 2 2 = 0), Q 2 q 2 1 Note that q 2 2 = 0, but q 2 0 for on-shell photon!

63 The LMD and LMD+V form factors Knecht, Nyffeler, EPJC 01; Nyffeler 09 Ansatz for VVP and thus F π 0 γ γ in large-n C QCD in chiral limit with 1 multiplet of lightest pseudoscalars (Goldstone bosons) and 2 multiplets of vector resonances, ρ, ρ (lowest meson dominance (LMD) + V) F π 0 γ γ fulfills all leading (and some subleading) QCD short-distance constraint from OPE Reproduces Brodsky-Lepage (BL): lim F π Q 2 0 γ γ (m2 π, Q 2, 0) 1/Q 2 (OPE and BL cannot be fulfilled simultaneously with only one vector resonance, LMD) Normalized to decay width Γ π 0 γγ LMD form factors (off-shell, on-shell, transition form factor): F LMD π 0 γ γ (q2 3, q 2 1, q 2 2) = Fπ 3 F LMD π 0 γ γ (q2 1, q 2 2) = Fπ 3 F LMD (Q 2 ) = Fπ 3M 2 V q q q 2 3 c V (q 2 1 M2 V ) (q2 2 M2 V ), q2 3 = (q 1 + q 2) 2 q q 2 2 c V (q 2 1 M2 V ) (q2 2 M2 V ), c V = c V m 2 π Q 2 + c V Q 2 + M 2 V Note that LMD transition form factor does not fall off like 1/Q 2 for large Q 2.

64 The LMD+V form factor Off-shell LMD+V form factor: F LMD+V π 0 γ γ (q 2 3, q 2 1, q 2 2) = Fπ 3 q 2 1 q 2 2 (q q q 2 3) + P V H (q 2 1, q 2 2, q 2 3) (q 2 1 M2 V 1 ) (q 2 1 M2 V 2 ) (q 2 2 M2 V 1 ) (q 2 2 M2 V 2 ) P V H (q 2 1, q 2 2, q 2 3) = h 1 (q q 2 2) 2 + h 2 q 2 1 q h 3 (q q 2 2) q h 4 q 4 3 +h 5 (q1 2 + q2) 2 + h 6 q3 2 + h 7 q3 2 = (q 1 + q 2) 2 F π = 92.4 MeV, M V1 = M ρ = MeV, M V2 = M ρ = GeV, Free parameters: h i On-shell LMD+V form factor: F LMD+V π 0 γ γ (q1, 2 q2) 2 = Fπ 3 q 2 1 q 2 2 (q q 2 2) + h 1 (q q 2 2) 2 + h 2 q 2 1 q h 5 (q q 2 2) + h 7 (q 2 1 M2 V 1 ) (q 2 1 M2 V 2 ) (q 2 2 M2 V 1 ) (q 2 2 M2 V 2 ) h 2 = h 2 + m 2 π, h 5 = h 5 + h 3m 2 π, h 7 = h 7 + h 6m 2 π + h 4m 4 π Transition form factor: F LMD+V (Q 2 ) = Fπ 3 1 M 2 V 1 M 2 V 2 h 1Q 4 h 5Q 2 + h 7 (Q 2 + M 2 V 1 )(Q 2 + M 2 V 2 ) h 1 = 0 in order to reproduce Brodsky-Lepage behavior. Can treat h 1 as free parameter to fit the BABAR data, but the form factor does then not vanish for Q 2, if h 1 0.

65 Fixing the LMD+V model parameters h i h 1, h 2, h 5, h 7 are quite well known: h 1 = 0 GeV 2 (Brodsky-Lepage behavior F LMD+V π 0 γ (m 2 γ π, Q 2, 0) 1/Q 2 ) h 2 = GeV 2 (Melnikov, Vainshtein 04: Higher twist corr. in OPE) h 5 = 6.93 ± 0.26 GeV 4 h 3 mπ 2 (fit to CLEO data of F LMD+V π 0 γ (m 2 γ π, Q2, 0)) h 7 = N C MV 4 1 MV 4 2 /(4π 2 Fπ) 2 h 6 mπ 2 h 4 mπ 4 = GeV 6 h 6 mπ 2 h 4 mπ 4 (or normalization to Γ(π 0 γγ)) Fit to BABAR data: h 1 = ( 0.17 ± 0.02) GeV 2, h 5 = (6.51 ± 0.20) GeV 4 h 3 mπ 2 with χ 2 /dof = 15.0/15 = 1.0. Result for a HLbL;π0 µ;lmd+v not affected (shifts compensate). h 3, h 4, h 6 are unknown / less constrained: New short-distance constraint h 1 + h 3 + h 4 = MV 2 1 MV 2 2 χ ( ) χ = quark condensate magnetic susceptibility of QCD (Ioffe, Smilga 84). LMD ansatz for VT χ LMD = 2/M 2 V = 3.3 GeV 2 (Balitsky, Yung 83) Close to χ(µ=1 GeV) = (3.15 ± 0.30) GeV 2 (Ball et al. 03) Assume large-n C (LMD/LMD+V) framework is self-consistent χ = (3.3 ± 1.1) GeV 2 vary h 3 = (0 ± 10) GeV 2 and determine h 4 from relation ( ) and vice versa Lattice QCD: χ MS u (µ = 2 GeV) = (2.08 ± 0.08) GeV 2 (Bali et al. 12). Final result for a LbyL;π0 µ is very sensitive to h 6 Assume LMD/LMD+V estimates of low-energy constants from chiral Lagrangian of odd intrinsic parity at O(p 6 ) are self-consistent. Assume 100% error on estimate for relevant low-energy constant h 6 = (5 ± 5) GeV 4

66 The VMD form factor Vector Meson Dominance: F VMD π 0 γ γ ((q1 + q2)2, q1, 2 q2) 2 = 12π 2 F π q1 2 M2 V q2 2 M2 V N C M 2 V M 2 V on-shell = off-shell form factor! Only two model parameters even for off-shell form factor: F π and M V Note: VMD form factor factorizes F VMD π 0 γ γ (q2 1, q 2 2) = f (q 2 1) f (q 2 2). This might be a too simplifying assumption / representation. VMD form factor has wrong short-distance behavior: F VMD π 0 γ γ (q2, q 2 ) 1/q 4, for large q 2, falls off too fast compared to OPE prediction F OPE π 0 γ γ (q2, q 2 ) 1/q 2. Transition form factor: N C M 2 V F VMD (Q 2 ) = 12π 2 F π Q 2 + MV 2

67 Anomalous magnetic moment in quantum field theory Quantized spin 1/2 particle interacting with external, classical electromagnetic field 4 form factors in vertex function (conserved electromagnetic current: j µ (x) = e ψ(x)γ µ ψ(x), momentum transfer k = p p, not assuming parity or charge conjugation invariance) γ(k) p p p, s j µ (0) p, s " = ( ie)ū(p, s ) γ µ F 1 (k 2 ) + iσµν k ν {z } 2m F 2(k 2 ) {z } Dirac Pauli +γ 5 σµν k ν 2m F 3(k 2 ) + γ 5 `k 2 γ µ k/ k µ F 4 (k 2 ) # u(p, s) Real form factors for spacelike k 2 0. Working out the low-momentum (non-relativistic) limits one obtains: F 1 (0) = 1 (renormalization of charge e) e µ = 2m (F 1(0) + F 2 (0)) (magnetic moment) F 2 (0) = a (anomalous magnetic moment) d = e 2m F 3(0) (electric dipole moment, violates CP) F 4 (k 2 ) = anapole moment (violates P)

68 Electron g 2: Theory Main contribution in Standard Model (SM) from mass-independent Feynman diagrams in QED with electrons in internal lines (perturbative series in α): a SM e = 5X α c n π n=1 n (2) [Loops in QED with µ, τ] (5) [weak interactions] (20) [strong interactions / hadrons] The numbers are based on the paper by Aoyama et al

69 QED: mass-independent contributions to a e α: 1-loop, 1 Feynman diagram; Schwinger 1947/48: c 1 = 1 2 α 2 : 2-loops, 7 Feynman diagrams; Petermann 1957, Sommerfield 1957: c 2 = π2 12 π2 2 ln ζ(3) = α 3 : 3-loops, 72 Feynman diagrams;..., Laporta & Remiddi 1996: c 3 = π π2 ln π2 ζ(3) ζ(5) = j Li ζ(3) 2160 π4 « ln π2 ln 2 2 α 4 : 4-loops, 891 Feynman diagrams; Kinoshita et al. 1999,..., Aoyama et al. 2008; 2012: c 4 = (20) (numerical evaluation) α 5 : 5-loops, Feynman diagrams; Aoyama et al. 2005,..., 2012: c 5 = 9.16(58) (numerical evaluation) Replaces earlier rough estimate c 5 = 0.0 ± 4.6. New result removes biggest theoretical uncertainty in a e! ff

70 Mass-independent 2-loop Feynman diagrams in a e 1) 2) 3) γ 4) 5) 6) µ γ µ γ e τ 7) 8) 7) 9)

71 Mass-independent 3-loop Feynman diagrams in a e 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27) 28) 29) 30) 31) 32) 33) 34) 35) 36) 37) 38) 39) 40) 41) 42) 43) 44) 45) 46) 47) 48) 49) 50) 51) 52) 53) 54) 55) 56) 57) 58) 59) 60) 61) 62) 63) 64) 65) 66) 67) 68) 69) 70) 71) 72)

72 Determination of fine-structure constant α from g 2 of electron Recent measurement of α via recoil-velocity of Rubidium atoms in atom interferometer (Bouchendira et al. 2011): α 1 (Rb) = (91) This leads to (Aoyama et al. 2012): ae SM (Rb) = (6) {z} (4) {z} (2) {z} c 4 c 5 had a exp e [0.66ppb] (78) [78] [0.67ppb] {z} α(rb) a SM e (Rb) = 1.09(0.83) [Error from α(rb) dominates!] Test of QED! Use ae exp to determine α from series expansion in QED (contributions from weak and strong interactions under control!). Assume: Standard Model correct, no New Physics (Aoyama et al. 2012): α 1 (a e) = (68) {z} (46) {z} (24) {z} c 4 c 5 had+ew (331) {z } a exp e [342] [0.25ppb] The uncertainty from theory has been improved by a factor 4.5 by Aoyama et al. 2012, the experimental uncertainty in ae exp is now the limiting factor. Today the most precise determination of the fine-structure constant α, a fundamental parameter of the Standard Model.

73 The Brookhaven Muon g 2 Experiment The first measurements of the anomalous magnetic moment of the muon were performed in 1960 at CERN, aµ exp = (14) (Garwin et al.) [12% precision] and improved until 1979: aµ exp = (85) [7 ppm] (Bailey et al.) In 1997, a new experiment started at the Brookhaven National Laboratory (BNL): µ Storage Ring Source: BNL Muon g 2 homepage spin momentum ω a = a µ eb mµ actual precession 2 Angular frequencies for cyclotron precession ω c and spin precession ω s: ω c = eb eb, ωs = m µ γ m µ γ + eb aµ, m µ ω a = a µ eb m µ γ = 1/ p 1 (v/c) 2. With an electric field to focus the muon beam one gets: ω a = e» a µ B a µ 1 «m µ γ 2 v E 1 Term with E drops out, if γ = p 1 + 1/a µ = 29.3: magic γ p µ = GeV/c

74 The Brookhaven Muon g 2 Experiment: storage ring Source: BNL Muon g 2 homepage

75 The Brookhaven Muon g 2 Experiment: result for a µ Million Events per 149.2ns Histogram with 3.6 billion decays of µ : Time modulo [µs] 100µs Source: Bennett et al N(t) = «t N 0 (E) exp γτ µ [1 + A(E) sin(ω at + φ(e))] Exponential decay with mean lifetime: τ µ,lab = γτ µ = µs (in lab system). Oscillations due to angular frequency ω a = a µeb/m µ. a µ = R λ R where R = ωa ω p and λ = µµ µ p Brookhaven experiment measures ω a and ω p (spin precession frequency for proton). λ from hyperfine splitting of muonium (µ + e ) (external input). With new CODATA 2008 value for λ, one obtains a new world average (dominated by BNL g 2 Experiment, Bennett et al. 2006): a exp µ = (63) [0.5 ppm]

76 Electron g 2: Experiment Latest experiment: Hanneke, Fogwell, Gabrielse, 2008 n = 2 ν c - 5δ/2 ν a n = 1 n = 2 ν c - 3δ/2 f c = ν c - 3δ/2 n = 1 n = 0 ν c - δ/2 ν a = gν c / 2 - ν c n = 0 m s = -1/2 m s = 1/2 Cylindrical Penning trap for single electron (1-electron quantum cyclotron) Source: Hanneke et al. g e 2 = νs ν a ν z /(2 f c) ν c f c + 3δ/2 + ν z 2 /(2 f + gcav c) 2 Cyclotron and spin precession levels of electron in Penning trap Source: Hanneke et al. ν s = spin precession frequency; ν c, ν c = cyclotron frequency: free electron, electron in Penning trap; δ/ν c = hν c/(m ec 2 ) 10 9 = relativistic correction 4 quantities are measured precisely in experiment: f c = ν c 3 δ 149 GHz; 2 νa = g νc νc 173 MHz; 2 ν z 200 MHz = oscillation frequency in axial direction; g cav = corrections due to oscillation modes in cavity a exp e = (28) [0.24 ppb 1 part in 4 billions] Only the 12th decimal place uncertain! (Kusch & Foley, 1947/48: 4% precision) Precision in g e/2 even 0.28 ppt 1 part in 4 trillions!