The muon g-2: where we are?

Transcription

1 The muon g-2: where we are? F. JEGERLEHNER Humboldt-Universität zu Berlin & DESY Zeuthen Final EURIDICE Collaboration Meeting Kazimierz, Poland, August 2006 supported by EU network EURIDICE and TARI

2 Outline of Talk: 1 The Anomalous Magnetic Moment of the Muon 2 Standard Model Prediction for a µ 3 Hadronic Light-by-Light Scattering Contribution to g 2 4 Outlook

3 1 The Anomalous Magnetic Moment of the Muon µ = g µ e 2m µ c s ; g µ = 2 (1 + a µ ) Dirac: g µ = 2, a µ muon anomaly Stern, Gerlach 22: g e = 2; Kusch, Foley 48: g e = 2 ( ± ) γ(q) µ(p ) µ(p) = ( ie) ū(p ) [ ] γ µ F 1 (q 2 ) + i σµν q ν 2m µ F 2 (q 2 ) u(p) F 1 (0) = 1 ; F 2 (0) = a µ a µ responsible for the Larmor precession directly proportional at magic energy 3.1 GeV CERN, BNL g-2 experiments ω a = e m [ a µ B ( ] E 3.1GeV a µ 1 β γ 1) E e 2 m at magic γ [ ] a µ B

4 The BNL muon storage ring

5 Measured: ratio R = ω s /ω c, ω s = precession frequency, ω c = cyclotron frequency; also need ratio µ µ /µ p between muon magnetic moment and proton magnetic moment a µ = R µ µ µ p R Distribution of counts versus time for the 3.6 billion decays in the 2001 negative muon data-taking period Million Events per 149.2ns Time modulo 100µs [µs]

6 µ - µ Avg. [τ] a µ Experiment + - [e e ] Theory a µ = (63) σ (e + e ) [ 1.4 σ (τ ) ] BNL - E821 Muon (g-2) Collaboration hep-ex/ / experiment 0.5ppm = [ stat, syst ] Uncertainties: theory 0.6ppm = (e + e ) [0.6ppm = (τ ) ] new BNL 969 prop. 0.2ppm = New physics sensitivity: (example) a SUSY µ /a µ 1.25ppm ( ) 2 100GeV m tan β m lightest SUSY particle; SUSY requires two Higgs dublets tan β = v 1 v 2, v i =< H i > ; i = 1, 2 χ χ µ µ tan β m t /m b 40 [4 40] ν χ 0 most NP models yield a µ (New Physics) m 2 µ /M 2 α π M 1 2 TeV

7 2 Standard Model Prediction for a µ QED Contribution The QED contribution to a µ has been computed (or estimated) through 5 loops Growing coefficients in the α/π expansion reflect the presence of large ln m µ m e 5.3 terms coming from electron loops. New: a e = (76) Gabrielse et al a QED α 1 (a e ) = (96) [0.70 ppb] µ = (0.03) (1.15) (0.08) }{{}}{{}}{{} α 4 α 5 α inp based on work of Kinoshita, Nio The current uncertainty is well below the ± experimental error from E821

8 # n of loops C i [(α/π) n ] a QED µ (0.44) (27) (0.02) (44) (0.00) (41) (1.19) (170) 6.29 (1.15) tot (2.80) ❶ 1 diagram Schwinger 1948 ❷ 7 diagrams Peterman 1957, Sommerfield 1957 ❸ 72 diagrams Lautrup, Peterman, de Rafael 1974, Laporta, Remiddi 1996 ❹ about 1000 diagrams Kinoshita 1999, Kinoshita, Nio 2004 ❺ estimate of leading terms Karshenboim 93, Czarnecki, Marciano 00, Kinoshita, Nio 05

9 Weak Contributions a weak(1) µ = (195 ± 0) Brodsky, Sullivan 67,..., W W ν µ + Z + H Bardeen, Gastmans, Lautrup 72 Higgs contribution tiny! a weak(2) µ = (44 ± 4) Kukhto et al 92 µ µ Z + γ W W ν µ ν µ W + e,u,d, Z γ + µ potentially large terms G F m 2 µ α π ln M Z m µ Peris, Perrottet, de Rafael 95 quark-lepton (triangle anomaly) partial Czarnecki, Krause, Marciano 96 Heinemeyer, Stöckinger, Weiglein 04, Gribouk, Czarnecki 05 final full 2 loop result known Most recent evaluations: improved hadronic part (beyond QPM) a weak µ = (152 ± 1[had]±?) ( Knecht, Peris, Perrottet, de Rafael 02) a weak µ = (154 ± 1[had] ± 2[m H, m t, 3 loop]) ( Czarnecki, Marciano, Vainshtein 02)

10 Hadronic Contributions General problem in electroweak precision physics: contributions from hadrons (quark loops) at low energy scales (a) e, µ, τ < γ Leptons γ > µ µ + γ γ(z) γ α : weak coupling pqed (b) γ γ µ (c) u,d, u,d, γ Z + µ γ + u, d, s, < γ Quarks g γ > α s : strong coupling pqcd (a) Hadronic vacuum polarization O(α 2 ), O(α 3 ) (b) Hadronic light-by-light scattering O(α 3 ) (c) Hadronic effects in 2-loop EWRC O(αG F m 2 µ ) Light quark loops Hadronic blob

11 Evaluation of a had µ Leading non-perturbative hadronic contributions a had µ can be obtained in terms of R γ (s) σ (0) (e + e γ hadrons)/ 4πα2 3s data via dispersion integral: a had µ = ( αmµ 3π ) ( E2 cut 2 + 4m 2 π E 2 cut ds Rdata γ (s) ˆK(s) s 2 ds RpQCD γ (s) ˆK(s) s 2 ) µ γ γ µ γ Data:...,CMD2,KLOE,SND Key role: VEPP-2M/Novosibirsk, DAFNE/Frascati Experimental error implies theoretical uncertainty! Low energy contributions enhanced: 67% of error on a had µ comes from region 4m 2 π < m2 ππ < M 2 Φ a had(1) µ = (692.0 ± 6.0) GeV ρ, ω 0.0 GeV, 3.1 GeV e + e data based φ, GeV

12 Novosibirsk Frascati battle fields after CMD-2 e + e (direct) contra ALEPH τ (indirect); KLOE (new technique: radiative return) clearly is on CMD-2 side (3% discrepancy vs 10 to 20%), SND joined, at the moment τ data not included since indirect and probably corrections not fully understood or experimental problem (Belle preliminary) Soon: BABAR and Belle will join the battle new KLOE result awaited!

13 F. Jegerlehner Final EURIDICE Meeting, Kazimierz, Poland August 24-27, 2006

14 Recent evaluations of a had µ data a had(1) µ Ref. hep-ph e + e 694.8[8.6] FJ,GJ e + e 692.0[6.0] FJ xxx e + e 696.3[7.2](6.2) exp (3.6) rad DEHZ e + e 690.9[4.4](3.9) exp (1.9) rad (0.7) QCD DEHZ 06 ICHEP06 e + e 692.4[6.4](5.9) exp (2.4) rad HMNT e + e 699.6[8.9](8.5) exp (1.9) rad (2.0) proc ELZ τ 711.0[5.8](5.0) exp (0.8) rad (2.8) SU(2) DEHZ e + e 693.5[5.9](5.0) exp (1.0) rad (3.0) l l TY e + e + τ 701.8[5.8](4.9) exp (1.0) rad (3.0) l l TY Differences in errors mainly by utilizing more or less theory: pqcd, SR, low energy QCD methods most recent data SND, CMD-2, BaBar included

15 Contributions to a had µ with relative (rel) and absolute (abs) error in percent. Energy range a had µ [%](error) 1010 rel [%] abs [%] ρ, ω (E < 2M K ) [ 77.8](3.84) M K < E < 2 GeV [ 14.8](4.72) GeV < E < M J/ψ [ 3.2](1.23) M J/ψ < E < M Υ [ 3.8](0.59) M Υ < E < E cut 1.40 [ 0.2](0.09) E cut < E pqcd 1.53 [ 0.2](0.00) E < E cut data [ 99.8](6.23) total [100.0](6.23)

16 % α (5) had (M2 Z ) GeV % α (5) had ( s 0) Comparison of error profiles between α (5) had (M 2 Z ) α (5) had ( s 0) and a µ GeV % a µ Note high improvement potential for VEPP-2000 and DANAE/KLOE GeV

17 Experimental Adler-function versus theory (pqcd + NP) in the low energy region ((EJKV98)). Note that the error includes both statistical and systematic ones, in contrast to R-data plots where only statistical errors are shown!!!. D γ (Q 2 ) = Q 2 4m 2 π ds R γ(s) (s+q 2 ) 2

18 Higher order hadronic contributions a had(2) µ = (100 ± 6) (Krause 96) a had(2) µ = ( 98 ± 1) (Hagiwara et al. 03) + +

19 3 Hadronic Light-by-Light Scattering Contribution to g 2 Hadronic light by light scattering a lbl µ = ( 80 ± 40) (Knecht&Nyffeler 02) a lbl µ = (136 ± 25) (Melnikov & Vainshtein 03) shift by (12) (13) (3) π 0, η, η π ±, K ± q = (u, d, s,...) L.D. L.D. S.D. Low energy effective theory: e.g. ENJL Kinoshita-Nizic-Okamoto 85, Hayakawa-Kinoshita-Sanda 95, Bijnens-Pallante-Prades 95

20 Hadrons in < 0 T {A µ (x 1 )A ν (x 2 )A ρ (x 3 )A σ (x 4 )} 0 > γ(k) k ρ had + 5 permutations of the q i q 3λ q 2ν q 1µ µ(p) µ(p ) Key object full rank-four hadronic vacuum polarization tensor Π µνλρ (q 1, q 2, q 3 ) = d 4 x 1 d 4 x 2 d 4 x 3 e i (q 1x 1 +q 2 x 2 +q 3 x 3 ) 0 T {j µ (x 1 )j ν (x 2 )j λ (x 3 )j ρ (0)} 0. non-perturbative physics general covariant decomposition involves 138 Lorentz structures of which 32 can contribute to g 2

21 fortunately, dominated by the pseudoscalar exchanges π 0, η, η,... described by the effective Wess-Zumino Lagrangian generally, pqcd useful to evaluate the short distance (S.D.) tail the dominant long distance (L.D.) part must be evaluated using some low energy effective model which includes the pseudoscalar Goldstone bosons as well as the vector mesons play key role u, d g u, d π 0 + π ± } {{ } L.D. } {{ } S.D. Hadronic light by light scattering is dominated by π 0 exchange in the odd parity channel, pion loops etc. at long distances (L.D.) and quark loops incl. hard gluonic corrections at short distances (S.D.)

22 The spectrum of invariant γγ masses obtained with the Crystal Ball detector. The three rather pronounced spikes seen are the γγ pseudoscalar (PS) γγ excitations: PS=π 0, η, η

23 Pion-pole contribution dominating hadronic contributions = neutral pion exchange γ π 0, η, η 1 µ 2 3 Leading hadronic light by light scattering diagrams The key object here is the π 0 γγ form factor F π 0 γ γ (m2 π, q1, 2 q2) 2 which is defined by the matrix element i d 4 xe iq x 0 T {j µ (x)j ν (0)} π 0 (p) = ε µναβ q α p β F π0 γ γ (m2 π, q 2, (p q) 2 ). Properties: bose symmetric F π 0 γ γ (s, q2 1, q2 2 ) = F π 0 γ γ (s, q2 2, q2 1 ) need it off-shell in integrals; if (s, q1, 2 q2) 2 (m 2 π, 0, q 2 ) in fact not known pion pole approximation in mind (pion pole dominance) s = m 2 π for generality use vertex function (not matrix element) i d 4 xe iq x 0 T {j µ (x) j ν (0) ϕ π 0(p)} 0 = ε µναβ q α p β F π 0 γ γ (p2, q 2, (p q) 2 ) i p 2 m 2 π,

24 with ϕ(p) = d 4 y e ipx ϕ(y) the Fourier transformed π 0 field. representation in terms of form factors To compute a LbL;π0 µ F M (0) pion pole, we need i k ρ Π (π0 ) µνλσ (q 1, q 2, k q 1 q 2 ) at k = 0 where p 3 = (p 1 + p 2 ). Computing the Dirac traces yields d a LbL;π0 µ = e 6 4 q 1 d 4 q 2 1 (2π) 4 (2π) 4 q1 2q2 2 (q 1 + q 2 ) 2 [(p + q 1 ) 2 m 2 ][(p q 2 ) 2 m 2 ] [ Fπ 0 γ γ (q2 2, q1, 2 q3) 2 F π 0 γ γ(q2, 2 q2, 2 0) T 1 (q 1, q 2 ; p) q 2 2 m2 π + F π 0 γ γ (q2 3, q2 1, q2 2 ) F π 0 γ γ(q 2 3, q2 3, 0) q 2 3 m2 π ] T 2 (q 1, q 2 ; p) where T 1 (q 1, q 2 ; p) and T 2 (q 1, q 2 ; p) are scalar kinematics factors; two terms unified by bose symmetry.,

25 The pion-photon-photon transition form factor Form factor function F π 0 γ γ (s, s 1, s 2 ) is largely unknown Fortunately some experimental data is available: The constant F π0 γγ(m 2 π, 0, 0) is well determined by the π 0 γγ decay rate The invariant matrix element reads (follows from Wess-Zumino Lagrangian)(f π F 0 ) M π 0 γγ = e 2 F π 0 γγ(0, 0, 0) = e2 N c 12π 2 f π = α πf π GeV 1, Information on F π0 γ γ(m 2 π, Q 2, 0) comes from experiments e + e e + e π 0 where the electron (positron) gets tagged to high Q 2 values. e (p b ) e (p t ) γ Q 2 > 0 π 0 γ q 2 0 e + e + Measurement of the π 0 form factor F π 0 γ γ(m 2 π, Q2, 0) at high space like Q 2

26 Data for F π 0 γ γ(m 2 π, Q2, 0) is available from CELLO and CLEO. Brodsky Lepage interpolating formula gives an acceptable fit to the data. F π0 γ γ(m 2 π, Q 2, 0) 1 4π 2 f π (Q 2 /8π 2 f 2 π ) Assuming the pole approximation this FF has been used by all authors (HKS,BPP,KN) in the past, but has been criticized recently (MV). In fact in g 2 we are at zero momentum such that only the FF F π 0 γ γ( Q 2, Q 2, 0) F π 0 γ γ(m 2 π, Q2, 0) is consistent with kinematics. Unfortunately, this off shell form factor is not known and in fact not measurable. An alternative way to look at the problem is to use the anomalous PCAC relation and to relate π 0 γγ to directly the ABJ anomaly. High energy behavior of F π 0 γ γ is best investigated by OPE of 0 T {j µ (x 1 )j ν (x 2 )j λ (x 3 )} γ(k) taking into account that the external photon is in a physical state. Looking at first diagram: q 1 and q 2 as independent loop integration momenta most important region q 2 1 q2 2 q2 3, short distance expansion of T {j µ(x 1 )j ν (x 2 )} for x 1 x 2.

27 x y x y = + Usual structure of OPE = x y + perturbative hard short distance coefficient function soft long distance matrix element Surprisingly, the leading possible term in is just given by the ABJ anomaly diagram known exact to all orders and given by the lowest order (one loop) result. This requires of course that the leading operator in the short distance expansion must involve the divergence axial current. Indeed, i d 4 x 1 d 4 x 2 e i (q 1x 1 +q 2 x 2 ) T {j µ (x 1 )j ν (x 2 )X} = d 4 z e i (q 1+q 2 ) z 2i ˆq 2 ε µναβ ˆq α T {j β 5 (z)x} + with j µ 5 = q ˆQ 2 γ µ γ 5 q the relevant axial current and ˆq = (q 1 q 2 )/2 q 1 q 2 ; the momentum flowing through the axial vertex is q 1 + q 2 and in the limit k 0 of our interest q 1 + q 2 q 3 which is assumed to be much smaller than ˆq (q 2 1 q 2 2 2q 3ˆq 0).

28 After the large q 1, q 2 behavior has been factored out the remaining soft matrix element to be calculated is T λβ = i d 4 z e i q3z < 0 T {j 5β (z)j λ (0)} γ(k) >, which is precisely the well known VVA triangle correlator T (a) λβ = i en c 4π 2 w L (q3) 2 q 3β q3 σ f σλ + w T (q3) ( q f λβ + q 3λ q3 σ f σβ q 3β q3 σ f ) σλ Both amplitudes the longitudinal w L as well as the transversal w T are calculable from the triangle fermion one loop diagram. In the chiral limit they are given by w (a) L (q2 ) = 2w (a) T (q2 ) = 2/q 2. as discovered by (Vainshtein 03). So everything is perturbative with no radiative corrections at all! At this stage of the consideration it looks like a real mystery what all this has to do with π 0 exchange, as everything looks perfectly controlled by perturbation theory. The clue is that as a low energy object we may evaluate this matrix element at the same time perfectly in terms of hadronic spectral functions by saturating it by a sum over intermediate states. Melnikov, Vainshtein: vertex with external photon must be non-dressed! i.e. no VDM damping result increases by 30%!

29 Basic problem: (s, s 1, s 2 ) domain of F π 0 γ γ (s, s 1, s 2 ); here (0, s 1, s 2 ) plane??? pqcd RχPT/ENJL??? Compare VP: s-axis RχPT/ENJL pqcd

30 Originally VMD model (Kinoshita et al 85): ρ mesons play an important role in the game (see hadronic VP) looks natural to apply a vector meson dominance (VDM) model Naive VDM i q 2 i q 2 m 2 ρ q 2 m 2 ρ = i q 2 i q 2 m 2 ρ. provides a damping at high energies, ρ mass as an effective cut-off (physical version of a Pauli-Villars cut-off) photons dressed photons! naive VDM violates electromagnetic WT-identity Correct implementation in accord with low energy symmetries of QCD: vector meson extended CHPT (EχPT) model (Ecker et al 89) hidden local symmetry (HLS) model (Bando et al 85 ) extended NJL (ENJL) model (Dhar et al 85, Ebert, Reinhardt 86, Bijnens 96) to large extend equivalent Problem: matching L.D. with S.D. results depend on matching cut off Λ model dependence (non-renormalizable low energy effective theory vs. renormalizable QCD) Novel approach: refer to quark hadron duality of large-n c QCD, hadrons spectrum known, infinite series of narrow spin 1 resonances ( t Hooft 79) no matching problem (resonance representation has to match quark

31 level representation) (De Rafael 94, Knecht, Nyffeler 02) lowest meson dominance (LMD) plus one vector state (V) approximation to large-n c QCD allows for correct matching LMD+V parametrization of π 0 γγ form factor (see below) Summary of most recent results a µ BPP HKS KN MV π 0, η, η 85 ± ± ± ± 10 π, K loops 19 ± ± ± 10 axial vector 2.5 ± ± ± 5 scalar 6.8 ± quark loops 21 ± ± total 83 ± ± ± ± 25 Note: MV and KN utilize the same model LMD+V form factor: F πγ γ (q2 1, q2 2 ) = 4π2 F 2 π N c q 2 1 q2 2 (q2 1 + q2 2 ) h 2q 2 1 q2 2 + h 5(q q2 2 ) + (N cm 4 1 M 4 2 /4π2 F 2 π ) (q M 2 1 )(q2 1 + M 2 2 )(q2 2 + M 2 1 )(q2 2 + M 2 2 ), where M 1 = 769 MeV, M 2 = 1465 MeV, h 5 = 6.93 GeV 4.

32 with two modifications: form factor F π 0 γ γ(q2 2, q2 2, 0) = 1: undressed soft photon (non-renormalization of ABJ) Note: to have anomaly correct doesn not imply that there is no damping! P V V anomaly quark loop is counter example; it has correct πγγ in chiral limit (anomaly) and goes like 1/q 2 i F π 0 γ γ (q2 2, q2 1, q2 3 ) F π 0 γ γ (m2 π, q2 1, q2 3 ) = KN up to logs in all directions with h 2 = 0 ± 20 GeV 2 (KN) vs. h 2 = 10 GeV 2 (MV) fixed by twist 4 in OPE (1/q 4 ) a 1 [f 1, f 1 ] different mixing scheme Criticism: KN Ansatz only covers (0, q 2 1, q 2 2) plane, with consistent kinematics depends on 3 variables 2 dim integral representation no longer valid. Is this the final answer? How to improve? A limitation to more precise g 2 tests? Looking for new ideas to get ride of model dependence In principle lattice QCD could provide an answer [far future ( yellow region only)] Ask STRING THEORY? (see below)

33 Summary of contributions: a µ in units 10 6, ordered according to their size (L.O. lowest order, H.O. higher order, LBL. light by light); (α 1 = ) L.O. universal (0) e loops (0) H.O. universal (0) L.O. hadronic (60) L.O. weak (0) H.O. hadronic (1) LBL. hadronic (40) τ loops (0) New Physics??? a exp µ a the µ = 277 ± [2.9 σ] H.O. weak (2) e+τ loops (0) theory (72) experiment (60)

34 $ * ' 2 0 * ' * ' 8 *+ (g 2)µ aµ Summary: experiment vs. theory! #" % &% + ( )( % 1&% % + ( % -./, '( )( )( 43 -./, < ; : 6 ( * 5+( ( % ( &% 7/ $ 8 5( ( 0 =?> % 6 ( &% / :- 66( 0 ( % % 5 8 ( *+ ( ' 0 ( C3 % % ' 8 ( *5''( LK JI H G &DFE Given theory results only differ by a had(1) µ! F. Jegerlehner Final EURIDICE Meeting, Kazimierz, Poland August 24-27, 2006

35 % $. #. /,!" % $ (g 2)µ aµ: type and size of contributions new physics µ ) aµ = a QED µ + a had(1) µ + a had(2) µ + a weak(1) µ + a weak(2) µ + a lbl µ (+a! ')( & $! * +-,! * & $ # All kind of physics meets! Almost all EURODAFNE & EURIDICE nodes involved in one or the other problem!!! F. Jegerlehner Final EURIDICE Meeting, Kazimierz, Poland August 24-27, 2006

36 4 Outlook BNL g 2 experiment 14 fold improvement (vs CERN 1979) to 0.54ppm SM Test: substantial improvement of CPT test a µ = (54)(33) confirms for the first time weak contribution: 2σ 3σ very sensitive to precise value of the hadronic contribution: limits theoretical precision 2σ 1σ, now δlbl Light-by-Light (model-dependent) not far from being as important as the weak contribution, may become the limiting factor for future progress (theory?) New Physics??? 2.9 [3.4] σ a exp µ a SM µ = (28 ± 9)[(27 ± 8)] Small discrepancy persists, however, established deviation from the SM, in contrary: very strong constraint on many NP scenarios

37 On prediction side: big experimental challenge: attempt cross-section measurements at 1% level up to J/ψ[Υ]!!! crucial for g 2 and α QED (M Z ) at GigaZ new ideas on how to get model independent hadronic LbL contribution theory of pion FF below 1 GeV allows for further improvement (Colangelo, Gasser, Leutwyler); constraints from ππ scattering phase shifts Plans for new g 2 experiment! Present: BNL E ppm Future: BNL E ppm J-PARC 0.1 ppm Hunting for precision goes on!!!

38 History of sensitivity to various contributions BNL CERN III CERN II CERN I 4th QED 6th 8th hadronic VP hadronic LBL weak New Physics??? SM precision a µ uncertainty [ppm]