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BNL-E821 04 + – 0 DEHZ 03 (E E -Based) Dispersion Integral, Ranging from the Π Γ Threshold to 180.9 ± 8.0 DEHZ 03 (Τ-Based) Infinity

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BNL-E821 04 + – 0 DEHZ 03 (E E -Based) Dispersion Integral, Ranging from the Π Γ Threshold to 180.9 ± 8.0 DEHZ 03 (Τ-Based) Infinity CPC(HEP & NP), 2010, 34(6): 718{727 Chinese Physics C Vol. 34, No. 6, Jun., 2010 New results on the hadronic vacuum polarization contribution to the muon g −2 Michel Davier1) Laboratoire de l'Acc´el´erateur Lin´eaire, IN2P3/CNRS, Universit´e Paris-Sud 11, Orsay 91898, France Abstract Results on the lowest-order hadronic vacuum polarization contribution to the muon magnetic anomaly are presented. They are based on the latest published experimental data used as input to the 0 + − + − dispersion integral. Thus recent results on τ ντππ decays from Belle and on e e annihilation to π π ! from BABAR and KLOE are included. The new data, together with improved isospin-breaking corrections for τ decays, result into a much better consistency among the different results. A discrepancy between the Standard Model prediction and the direct g 2 measurement is found at the level of 3σ. − Key words muon magnetic moment, vacuum polarization, electron-positron annihilation, tau decays, g 2 − PACS 13.40Em, 13.60.Hb, 13.66.Bc 1 Introduction is on a level with the τ data in that channel, sys- tematic discrepancies in shape and normalisation of The Standard Model (SM) prediction of the the spectral functions were observed between the two anomalous magnetic moment of the muon, aµ, is lim- systems [6, 7]. It was found that, when computing ited in precision by contributions from hadronic vac- the hadronic VP contribution to the muon magnetic uum polarisation (HVP) loops. These contributions anomaly using the τ instead of the e+e− data for the can be conveniently separated into a dominant low- 2π and 4π channels, the observed deviation with the had;LO had;HO est order (aµ ) and higher order (aµ ) parts. experimental value would reduce to less than 1σ [1]. The lowest order term can be calculated with a com- Fig. 1 summarizes the comparison between theory bination of experimental cross section data involv- and experiment by 2006-8 [1]. ing e+e− annihilation to hadrons, and perturbative QCD. These are used to evaluate an energy-squared BNL-E821 04 + ± 0 DEHZ 03 (e e -based) dispersion integral, ranging from the π γ threshold to 180.9 ± 8.0 DEHZ 03 (τ-based) infinity. The integration kernel strongly emphasises 195.6 ± 6.8 the low-energy part of the spectrum, dominated by HMNT 03 (e+e±-based) 176.3 ± 7.4 2) + − the ππ final state . When using e e data a devia- J 03 (e+e±-based) tion of more than 3σ was observed [1{3] between the 179.4 ± 9.3 (preliminary) TY 04 (e+e±-based) SM prediction and the direct experimental value [4]. 180.6 ± 5.9 (preliminary) + ± + − DEHZ ICHEP 2006 (e e -based) A former lack of precise e e -annihilation data in- 180.5 ± 5.6 (preliminary) spired the search for an alternative. It was found [5] 208 ± 5.8 BNL-E821 04 in form of τ ν +hadrons spectral functions, trans- 208 ± 5.8 ! τ ferred from the charged to the neutral state using 140 150 160 170 180 190 200 210 isospin symmetry. During the last decade, new mea- ±10 aµ ± 11 659 000 (10 ) + − surements of the ππ spectral function in e e an- Fig. 1. Comparison of the predictions for the nihilation with percent accuracy became available, muon magnetic anomaly with the BNL mea- superseding or complementing older and less precise surement [4] in 2006, from (top to bottom) data. With the increasing precision, which today Refs. [1, 7{10]. Received 25 January 2010 1) E-mail: [email protected] 2) Throughout this paper, final state photon radiation is implied for hadronic final states ©2009 Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Sciences and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd No. 6 Michel Davier: New results on the hadronic vacuum polarization contribution to the muon g 2 719 − In this review I present the situation as of Oc- the risk of double-counting some of the higher-order tober 2009, taking advantage of very recent papers: contributions. (1) an updated analysis [11] using τ data, includ- The function K(s) 1=s in Eq. (3) gives a strong ∼ ing high-statistics Belle results [12] and an improved weight to the low-energy part of the integral. About had;LO treatment of isospin-breaking corrections (IB) [13]; 91% of the total contribution to aµ is accumu- (2) a BABAR measurement [14] of the ππ spectral lated at centre-of-mass energies ps below 1.8 GeV function using the hard initial state radiation (ISR) and 73% is covered by the ππ final state, which is method, benefiting from a large cancellation of sys- dominated by the ρ(770) resonance. tematic effects in the ratio ππγ(γ) to µµγ(γ) em- ployed for the measurement; and (3) a global analy- 3 Updated 2π analysis using τ data sis [15] of all published e+e−data. 3.1 Spectral functions in τ decays 2 HVP and g −2 The spectral function of the vector current decay τ X−ν is related to the e+e− X0 cross section It is convenient to separate the Standard Model ! τ ! of the corresponding isovector final state X0, (SM) prediction for the anomalous magnetic moment 2 I=1 4πα of the muon into different contributions, σ 0 (s) = v − (s) ; (4) X s 1; X SM QED had weak aµ = aµ +aµ +aµ ; (1) where s is the centre-of-mass energy-squared or equiv- with alently the invariant mass-squared of the τ final state had had;LO had;HO had;LBL X, α is the electromagnetic fine structure constant, aµ = aµ +aµ +aµ ; (2) and v1; X− is the non-strange, isospin-one vector spec- QED −10 and where aµ = (11658471:810 0:016) 10 is the tral function corrected for IB and given by had;LO pure electromagnetic contribution [16], aµ is the 2 mτ X− 1 dNX had;HO v1; X− (s) = B lowest-order HVP contribution, aµ = ( 9:79 2 − − 6 Vud e NX ds × 0:08 0:03 ) 10 10 is the corresponding higher- j j B exp rad −2 −1 weak −10 s 2s RIB(s) order part [8, 17], and aµ = (15:4 0:1 0:2) 10 , 1 1+ ; (5) 2 2 where the first error is the hadronic uncertainty and − mτ mτ SEW the second is due to the Higgs mass range, accounts with 3 2 for corrections due to exchange of the weakly inter- FSR(s) β0 (s) F0(s) RIB(s) = 3 : (6) G (s) β−(s) F−(s) acting bosons up to two loops [18]. For the light-by- EM had;LBL In Eq. (5), (1=N )dN =ds is the normalised invari- light (LBL) scattering part, aµ , we use the value X X (10:5 2:6) 10−10 from the latest evaluation [19]. ant mass spectrum of the hadronic final state, and − Owing to unitarity and to the analyticity of the X− denotes the branching fraction of τ ντX . B ! vacuum-polarization function, the lowest order HVP We use for the τ mass the value mτ = (1776:84 contribution to aµ can be computed through the dis- 0:17) MeV [21], and for the CKM matrix element persion integral [20] Vud = 0:97418 0:00019 [22], which assumes CKM j j 2 1 unitarity. For the electron branching fraction we use had;LO α K(s) (0) aµ = 2 ds R (s) ; (3) e = (17:818 0:032)%, obtained [23] supposing lep- Z 2 3π 4mπ s B ton universality. Short-distance electroweak radia- where K(s) is a well-known QED kernel, α = α(s = 0), tive effects lead to the correction SEW = 1:0235 (0) and R (s) denotes the ratio of the \bare" cross sec- 0:0003 [6, 24{26]. All the s-dependent IB corrections tion for e+e− annihilation into hadrons to the point- are included in RIB, which have been worked out in like muon-pair cross section. The bare cross section the 2π channel: FSR(s) refers to the final state ra- is defined as the measured cross section corrected for diative corrections [27], and GEM(s) denotes the long- initial-state radiation, electron-vertex loop contribu- distance radiative corrections of order α to the pho- tions and vacuum-polarization effects in the photon ton inclusive τ spectrum, computing the virtual and propagator. However, photon radiation in the final real photonic corrections using chiral resonance [28] state is included in the bare cross section defined or vector dominance [29]. here. The reason for using the bare (i.e., lowest or- der) cross section is that a full treatment of higher 3.2 Improved IB corrections orders is anyhow needed at the level of aµ, so that As the physics of IB is described elsewhere [13] I the use of the \dressed" cross section would entail only concentrate here on the major difference between 720 Chinese Physics C (HEP & NP) Vol. 34 the previous analysis [6, 7] and the new one [11]. In shape of their mass dependence can be compared by Eq. (6) the ratio of the electromagnetic F0(s) and looking at the relative difference between each spec- weak F−(s) form factors depends on the charged tral function and the combined one, locally averaging and neutral ρ parameters, as well as ρ-! interfer- the data from ALEPH [33], CLEO [34], OPAL [35], ence. While a small mass difference (of the order of and Belle [12] (Fig.
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