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’Accélérateur Linéaire, IN2P3/CNRS, Université 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, systematic 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 √s 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- | | 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 | | 2 ∞ 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 order) 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-ω interfer- the data from ALEPH [33], CLEO [34], OPAL [35], ence. While a small mass difference (of the order of and Belle [12] (Fig. 2). 1 MeV) makes only a small effect on the dispersion had,LO integral because of its bipolar nature, a difference in Table 1. Contributions to aµ [ππ,τ] from the width can lead to a significant correction. Such a the isospin-breaking corrections. Corrections difference is expected from radiative ρ ππγ decays shown in two separate columns correspond → which have been evaluated in a scalar-QED vector- to the Gounaris-Sakurai (GS) and Kuhn-¨ Santamaria (KS) form factor parametrisa- dominance model [30]. The result is markedly differ- tions, respectively. ent from the estimate made previously using only the had,LO −10 hard radiation part [31]. ∆ aµ [ππ,τ] (10 ) source One could question the validity of using point-like GS Model KS Model S 12.21 0.15 pions in the calculation of radiative decays. How- EW − G 1.92 0.90 ever several experimental tests support this assump- EM − + − + − FSR +4.67 0.47 tion in e e π π γ(γ) for the same mass range: → ρ-ω interference +2.80 0.19 +2.80 0.15 lowest-order FSR with KLOE [32], additional FSR m m 0 (σ) 7.88 with BABAR [14]. π − π − mπ mπ0 (Γρ) +4.09 +4.02 The new IB corrections are listed in Table 1. − +0.27 +0.19 mρ mρ0 +0.20− . +0.11− . − bare 0 19 0 11 3.3 Consistency of τ spectral functions ππγ, EM decays 5.91 0.59 6.39 0.64 − − All published τ 2π spectral functions are nor- total 16.07 0.59 16.70 0.64 − − 16.07 1.85 malised to the world-average branching ratio. The −

0.3 0.3 ALEPH CLEO 0.2 0.2 Combined (A-C-O-B) Combined (A-C-O-B) 0.1 0.1 0 0 Exp/Combined-1 Exp/Combined-1 -0.1 -0.1 -0.2 -0.2 -0.3 -0.3 0.2 0.4 0.6 0.8 1 1.2 0.2 0.4 0.6 0.8 1 1.2 m2(ππ0) (GeV2) m2(ππ0) (GeV2) 0.3 0.3 OPAL Belle 0.2 0.2 Combined Combined (A-C-O-B) 0.1 (A-C-O-B) 0.1 0 0 Exp/Combined-1 Exp/Combined-1 -0.1 -0.1 -0.2 -0.2 -0.3 -0.3 0.2 0.4 0.6 0.8 1 1.2 0.2 0.4 0.6 0.8 1 1.2 m2(ππ0) (GeV2) m2(ππ0) (GeV2)

0 Fig. 2. Relative comparison between the τ ντππ spectral functions from ALEPH, CLEO, OPAL, Belle → (data points) and their combined result (shaded band).

Since the world-average branching ratio is domi- a very good agreement between the full dispersion in- nated by the ALEPH result, it is interesting to test tegrals with comparable uncertainties. Thus the τ the consistency between the absolute spectra, i.e. experiments yield consistent absolute results. The when each spectrum is normalised to the branching average value and its error are rather insensitive to ratio measured by the same experiment. Fig. 3 shows the branching ratio choice, although it is not true at No. 6 Michel Davier: New results on the hadronic vacuum polarization contribution to the muon g 2 721 −

0.3 -1 the level of individual experiments. In particular the 2 | τ CMD2 03-06 SND 06 /|F

Belle result reaches its best precision only when the 2 0.2 | Combined τ (A-C-O-B) ee world-average is used. |F 0.1

-0.1

-0.2 +δ(IB corrections) -0.3 0.2 0.4 0.6 0.8 1 1.2 s (GeV2) 0.3 -1 2 | τ KLOE 08 /|F

2 0.2 | Combined τ (A-C-O-B) ee |F 0.1

-0.1

-0.2 +δ(IB corrections) -0.3 0.2 0.4 0.6 0.8 1 1.2 s (GeV2) Fig. 4. Relative comparison between ee and τ had,LO Fig. 3. Comparison of the aµ [ππ] values spectral functions, expressed in terms of the for diﬀerent τ experiments using their own diﬀerence between neutral and charged pion 0 τ ντππ branching ratios (closed circles) or → form factors. Isospin-breaking corrections are the world-average (open circles). applied to τ data with the corresponding uncertainties included in the error band. 3.4 Comparison to e+e− Data

τ decays Belle Figure 4 shows the relative diﬀerence between the 25.24 ± 0.01 ± 0.39 CLEO ee and the IB-corrected τ spectral functions versus s. 25.44 ± 0.12 ± 0.42 ALEPH The relative normalisation is consistent within the re- 25.49 ± 0.10 ± 0.09 DELPHI 25.31 ± 0.20 ± 0.14 spective errors and the shape is found in better agree- L3 24.62 ± 0.35 ± 0.50 ment than before [7], despite a remaining deviation OPAL 25.46 ± 0.17 ± 0.29 above the ρ-mass-squared. The discrepancy with the τ average 25.42 ± 0.10 KLOE data, although reduced, persists. e+e– CVC CMD2 03 As the g 2 dispersion relation involves an integral 25.03 ± 0.29 − CMD2 06 over the hadronic spectral function, it is interesting to 24.94 ± 0.31 SND 06 consider the result with another kernel. By integrat- 24.90 ± 0.36 KLOE 08 + − 24.64 ± 0.29 ing the e e data weighted by the τ matrix element e+e− average 24.78 ± 0.28 Cτ, and correcting for IB, one obtains the branching CVC 23.5 24 24.5 25 25.5 26 26.5 27 ratio Bππ0 which can be directly compared to the B(τ– → ν π–π0) (%) measurements. Indeed, τ

2 Fig. 5. The measured branching fractions for 2 mτ 3 SEWBe Vud σπ+π− 0 CVC τ ντππ (references in [11]) compared to the B 0 = dss C , (7) ππ 2 | 2 | τ → − − Z 2 + + 2 πα mτ 4mπ RIB predictions from the e e π π spectral → 2 functions, applying the IB corrections. For s 2s C = 1 1+ . (8) the ee results, only the data from the indicated τ m2 m2 − τ τ experiments in the 0.63–0.958 GeV range are CVC + − The results for Bππ0 from e e experiments are used, and the combined ee data elsewhere. compared to the direct measurements in Fig. 5 The Vertical bands indicate average values. average, (24.78 0.17 0.22 )%, differ from the exp IB average τ branching ratio, (25.42 0.10)%, by (0.64 2σ is significantly reduced from the previous analy- 0.10 0.17 0.22 )% to be compared to an applied sis [1] (4.5σ). It should be emphasized that the ob- τ ee IB IB correction of (+0.69)%. The discrepancy of about served increased deviation above the ρ mass between 722 Chinese Physics C (HEP & NP) Vol. 34 ee and τ spectral functions, essentially driven by the erage and measure the remaining systematic error, CVC KLOE data, plays a more significant role in the Bππ0 and (iii) minimise the integral error after averaging integral, rather than in the aµ integral with its much while respecting the two previous requirements. The steeper kernel. So we expect a better consistency for first item practically requires the use of pseudo-Monte g 2. Carlo (MC) simulation, which needs to be a faithful − representation of the measurement ensemble and to + 4 Updated 2π analysis using e e− in- contain the full data treatment chain (interpolation, cluding BABAR 4

4.1 The data 3.5

3 Recent precision data, where all required radiative Error scale factor corrections have been applied by the experiments, 2.5 stem from the CMD-2 [37] and SND [38] experiments at the VEPP-2M collider. They achieve comparable 2 statistical errors, and energy-dependent systematic uncertainties down to 0.8% and 1.3%, respectively. 1.5

These measurements have been complemented by 1 φ 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 results from KLOE [39] at DAΦNE running at the s [GeV] resonance centre-of-mass energy. KLOE applied for the first time the ISR technique to precisely determine 1 the ππ cross section between 0.592 and 0.975 GeV. 0.8 The high statistics of the analysed data sample yields BABAR KLOE a 0.2% relative statistical error on the ππ contribution 0.6 SND CMD2 (2004) had,LO to aµ . KLOE normalises the ππγ cross section Relative weight in average CMD2 (2006) Other exp taking the absolute ISR radiator function from Monte 0.4 Carlo simulation (Ref. [40] and references therein). The systematic error assigned to this correction varies 0.2 φ between 0.5% and 0.9% (closer to the peak). The 0 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 total assigned systematic error lies between 0.8% and s [GeV] 1.2%. In a recent publication [14] the BABAR Col- laboration reported measurements of the processes ee ππγ,µµγ using the ISR method at 10.6 GeV → centre-of-mass energy. The detection of the hard ISR photon allows BABAR to cover a large energy range from threshold up to 3 GeV for the two processes. The ππ(γ) cross section is obtained from the ππγ(γ) to µµγ(γ) ratio, so that the ISR radiation function cancels, as well as additional ISR radiative effects. Since additional FSR photons are also detected, there is no additional uncertainty from radiative corrections at NLO level. Experimental systematic uncertainties Fig. 6. (color online) Top: Error rescaling fac- are kept to 0.5% in the ρ peak region (0.6–0.9 GeV), tor accounting for inconsistencies among experiments versus √s. Middle: Relative aver- increasing to 1% outside. aging weights for each experiment versus √s. 4.2 Combining cross section data Bottom: Contribution to the dispersion integral for the combined ee data obtained by The details of the combination procedure are multiplying the ππ cross section by the kernel given in Ref. [15]. The requirements for averaging function K(s) (solid line). The dashed (red) and integrating cross section data are: (i) properly curve belonging to the right axis shows the propagate all the uncertainties in the data to the corresponding error contribution, where sta- final integral error, (ii) minimise biases, i.e., repro- tistical and systematic errors have been added duce the true integral as closely as possible in av- in quadrature. No. 6 Michel Davier: New results on the hadronic vacuum polarization contribution to the muon g 2 723 − averaging, integration). The second item requires polation function within a bin is renormalised to keep a flexible data interpolation method and a realistic the integral in that bin invariant after the interpola- truth model used to test the accuracy of the integral tion. The final interpolation function per experiment computation with pseudo-MC experiments. Finally, within its applicable energy domain is discretised into the third item requires optimal data averaging taking small (1 MeV) bins for the purpose of averaging and into account all known correlations to minimise the numerical integration. spread in the integral measured from the pseudo-MC The averaging of the interpolated measurements sample. from different experiments contributing to a given en- The combination and integration of the ee ππ ergy bin is the most delicate step in the analysis chain. → cross section data is performed using the newly de- Correlations between measurements and experiments veloped software package HVPTools [36]. It trans- must be taken into account. Moreover, the exper- forms the bare cross section data and associated sta- iments have different measurement densities or bin tistical and systematic covariance matrices into fine- widths within a given energy interval and one must grained energy bins, taking into account to our best avoid that missing information in case of a lower mea- knowledge the correlations within each experiment as surement density is substituted by extrapolated infor- well as between the experiments (such as uncertain- mation from the polynomial interpolation. To derive ties in radiative corrections). The covariance matri- proper weights given to each experiment, wider av- ces are obtained by assuming common systematic er- eraging regions are defined to ensure that all locally ror sources to be fully correlated. To these matrices available experiments contribute to the averaging re- are added statistical covariances, present for exam- gion, and that in case of binned measurements at ple in binned measurements as provided by KLOE, least one full bin is contained in it. The averaging BABAR or the τ data, which are subject to bin-to- regions are used to compute weights for each exper- bin migration that has been unfolded by the experiment, which are applied in the bin-wise average of iments, thus introducing correlations. The interpo- the original finely binned interpolation functions. If lation between adjacent measurements of a given ex- the χ2 value exceeds the number of degrees of free- periment uses second-order polynomials, which is an dom (ndof ), the error in the averaged bin is rescaled 2 improvement with respect to the previously applied by χ /ndof to account for inconsistencies. Fig. 6 trapezoidal rule. In the case of binned data, the inter- showsp the distributions in √s of the error recaling

0.2 0.2 Average Average BABAR KLOE 0.15 0.15

0.1 0.1

0.05 0.05

-0 -0

-0.05 -0.05 Cross section(exp) / Average - 1 Cross section(exp) / Average - 1

-0.1 -0.1

-0.15 -0.15

-0.2 -0.2 0.5 0.6 0.7 0.8 0.9 1 0.5 0.6 0.7 0.8 0.9 1 s [GeV] s [GeV] 0.2 0.2 Average Average CMD2 (2006) SND 0.15 0.15 CMD2 (2004)

0.1 0.1

0.05 0.05

-0 -0

-0.05 -0.05 Cross section(exp) / Average - 1 Cross section(exp) / Average - 1

-0.1 -0.1

-0.15 -0.15

-0.2 -0.2 0.5 0.6 0.7 0.8 0.9 1 0.5 0.6 0.7 0.8 0.9 1 s [GeV] s [GeV] Fig. 7. Relative cross section comparison between individual experiments (symbols) and the HVPTools average (shaded band) computed from all measurements considered. Shown are BABAR (top left), KLOE (top right), CMD-2 (bottom left) and SND (bottom right). 724 Chinese Physics C (HEP & NP) Vol. 34

had,LO factor, the relative weights for each experiment, and Table 2. Evaluated aµ [ππ] contributions the contribution to the dispersion integral, as well as from the ee data for different energy intervals its error. It is seen that BABAR dominates the av- and experiments. Where two errors are given, eraging up to the ρ peak and above 0.95 GeV, while the first is statistical and the second system- KLOE has a larger weight in-between owing to the atic. The last value in parentheses is the total error). Also given is the τ-based result. steep behaviour of the radiator function when ap- had,LO −10 proaching 1 GeV. The uncertainty in the integral is √s/GeV Exp. aµ [ππ] (10 ) dominated by the measurements below 0.8 GeV. 2m .3 ee fit 0.55 0.01 π − The consistent propagation of all errors into the 0.30 0.63 Comb. ee 132.6 0.8 1.0 (1.3) − evaluation of ahad,LO is ensured by generating large 0.63 0.958 CMD2 03 361.8 2.4 2.1 (3.2) µ − samples of pseudo experiments, representing the full CMD2 06 360.2 1.8 2.8 (3.3) list of available measurements and taking into ac- SND 06 360.7 1.4 4.7 (4.9) KLOE 08 356.8 0.4 3.1 (3.1) count all known correlations. For each generated BABAR 365.2 1.9 1.9 (2.7) set of pseudo measurements, the identical interpo- Comb. ee 360.8 0.9 1.8 (2.0) lation and averaging treatment leading to the com- 0.958 1.8 Comb. ee 14.4 0.1 0.1 (0.2) putation of Eq. (3) as for real data is performed, − total Comb. ee 508.4 1.3 2.6 (2.9) hence resulting in a probability density distribution had,LO total Comb. τ 515.2 2.0 2.7 (3.4) for aµ [ππ], the mean and RMS of which define the 1σ allowed interval. The fidelity of the full analysis chain (polyno- 5 Multihadronic contributions mial interpolation, averaging, integration) has been tested with toy models, using as truth representation We also reevaluate the e+e− π+π−2π0 contri- had,LO → a Gounaris-Sakurai vector-meson resonance model bution to aµ . It is found that the CMD-2 data faithfully describing the ππ data. Negligible biases used previously have been superseded by modified or below 0.1 (10−10 units) are found, increasing to 0.5 more recent, but yet unpublished data [41], recovering (1.2 without the high-density BABAR data) when agreement with the published SND cross sections [42]. using the trapezoidal rule for interpolation instead of Since the new data are unavailable, we discard the second order polynomials. obsolete CMD-2 data from the ππ2π0 average, find- The relative differences between BABAR KLOE, ing ahad,LO[ππ2π0]= 17.6 0.4 1.7 (compared µ stat syst CMD-2, SND, and their average are given in Fig. 7. to 17.0 0.4 1.6 when including the obsolete stat syst Fair agreement is observed, though with a tendency CMD-2 data). The corresponding cross section mea- to larger (smaller) cross sections above 0.8 GeV for surements and HVPTools average are shown in Fig. 8. ∼ BABAR (KLOE). These inconsistencies (among oth- From the still preliminary BABAR results [43] it is ers) lead to the error rescaling shown in Fig. 6.

had,LO ππ ND 4.3 Results for aµ [ ] 50 M3N DM2 had,LO OLYA A compilation of results for aµ [ππ] for the var- 40 SND ious sets of experiments and energy regions is given Average Cross section [nb] in Table 2. The inclusion of the new BABAR data 30 signiﬁcantly increases the central value of the inte- 20 gral, without however providing a large error reduc-

tion, because of the incompatibility between mainly 10 BABAR and KLOE, causing an increase of the com-

0 bined error. In the energy interval between 0.63 and 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 had,LO 0.958 GeV, the discrepancy between the aµ [ππ] s [GeV] − evaluations from KLOE and BABAR amounts to Fig. 8. Cross section measurements for e+e − → 2.0σ. π+π 2π0 used in the calculation of had,LO 0 Since BABAR is the only experiment covering aµ [ππ2π ]. The shaded band depicts

the entire energy region between 2mπ and 1.8 GeV, the HVPTools interpolated average within had,LO it can provide its own evaluation [14] of aµ [ππ], 1σ errors. The individual measurements are 514.1 2.2 3.1 1). referenced in Ref. [6]. stat syst

−10 1) When not speciﬁed, the aµ values are given in units of 10 . No. 6 Michel Davier: New results on the hadronic vacuum polarization contribution to the muon g 2 725 −

clear that the region above 1.4 GeV is still under- A compilation of recent SM predictions for aµ estimated at the present state, as corroborated by τ compared with the experimental result is given in data [6]. Fig. 9. The BABAR results are not yet contained in previous evaluations. The result by HMNT [2] con- 6 Results and comparison to experi- tains older KLOE data [45], which have been super- ment seded by more recent results [39] leading to a slightly had,LO larger value for aµ .

had,LO Adding to the ee-based aµ [ππ] and had,LO 0 aµ [ππ2π ] results the remaining exclusive multi- 7 Conclusions: discussion and perpec- hadron channels as well as perturbative QCD [1], we tives find for the complete lowest-order hadronic term had,LO The following concluding remarks can be made: aµ [ee] = 695.5 4.0exp 0.7QCD (4.1tot). (9) 1) The first point to emphasize is the better con- It is noticeable that the error from the ππ channel sistency between the ee and τ analyses, resulting from now equals the one from all other contributions to the improved IB corrections and the BABAR results. had,LO aµ . There is still a difference of (6.8 2.9ee 3.4τ) (1.5σ) in Adding further the other contributions (given in the ππ channel, but it can be considered as reason- Section 2), we obtain the Standard Model prediction able. The discrepancy was 2.9σ before, reduced to (still in 10−10 units) 2.4σ after the τ update. The other major difference aSM [ee] = 11659183.4 4.1 2.6 0.2 (4.9 ), (10) affecting the two estimates is in the 2π2π0 channel, µ tot (3.8 1.7ee 1.4τ) (1.7σ). For this case we have seen where the errors have been split into lowest and − that a better measurement with e+e will likely move higher-order hadronic, and other contributions, re- the ee result closer to the τ value. spectively. The aSM [ee] value deviates from the ex- µ 2) The internal consistency of the ee ππ data perimental average [4], aexp = 11659208.9 5.4 3.31), → µ is only fair, as a discrepancy is observed between the by 25.5 8.0 (3.2σ). For comparison the difference ob- BABAR and KLOE results, which is increasing with tained with the updated τ analysis is 15.7 8.2 (1.9σ). energy. The difference on the ρ peak, about 3% (be- yond the respective systematic errors of 0.5% and BNL-E821 2004

HMNT 07 (e+e–) 1.1%), is the most damaging for the dispersion inte- –276 ± 51 gral. Here CMD-2 agrees well with BABAR , while JN 09 (e+e–) –290 ± 65 SND lies between BABAR and KLOE. However, the Davier et al. 09 (τ) accuracy of both CMD-2 and SND is not enough to

–148 ± 52 resolve the issue. Davier et al. 09 (e+e–) –303 ± 51 3) While the BABAR measurement is explicitly Davier et al. 09 (e+e– w/ BABAR) done at NLO (including one additional ISR or FSR –246 ± 49 photon), it is insensitive to the Monte Carlo NLO gen- BNL-E821 (WA) eration. The situation is diﬀerent for KLOE which 0 ± 63 relies on Phokhara for the ISR radiation function. -600 -500 -400 -300 -200 -100 0 100 –11 Using the ISR process ee µµ BABAR has been exp × 10 → aµ – aµ able to verify that Phokhara provides the right an- SM Fig. 9. Compilation of recent results for aµ , swer to an accuracy of 1.1%, however in a kinematic subtracted by the central value of the experi- region (very hard ISR photons, x = 1 s/s > 0.9, − 0 mental average [4]. The shaded vertical band where s0 is the square of the ee CM energy) far from indicates the experimental error. The SM pre- that of KLOE (0.09 < x < 0.66). A measurement of dictions are taken from: HMNT 07 [2], JN the muon ISR process by KLOE has been considered τ 09 [46], Davier et al. 09 [11] ( -based and ee since some time and would be of considerable help to before BABAR ), and the ee-based value [15] validate their approach. including BABAR . 4) Although the accuracy of the BABAR results

1)The g 2 measurement is obtained from the ratio of two frequencies and needs as input the ratio of the muon to the proton − magnetic moments. The latter ratio is derived from muonium hyperfine splitting, and its value has been updated [44] after the −10 E-821 publication. The new value produces a shift of +0.92 10 of the aµvalue. 726 Chinese Physics C (HEP & NP) Vol. 34 is similar to that of the combined previous ee exper- higher-energy VEPP-2000 collider [47]. iments, the gain in precision that could have been 8) The theory error is still dominated by the un- hoped for was not realised because of the remaining certainty on the HVP contribution (4.1)1), but it is discrepancy, essentially with KLOE. now close to that of the hadronic LBL part (2.6)1). 5) All approaches now yield a deviation from the Since the latter contribution is unlikely to be known direct measurement, however at different levels de- more precisely in the short-term it will eventually be pending on what ππ input data are used: 2.4σ with the theory show-stopper. BABAR alone, 3.2σ with all ee data, 3.7σ with ee 9) But the real limitation at the moment is the not including BABAR , 2.9σ with ee not including g 2 measurement itself. The uncertainty reached by − KLOE, and 1.9σ with τ alone. E-821 is 6.3, larger than the full theory error (4.9). 6) Considering these results one can say there is It is therefore mandatory to pursue these measure- some evidence for a deviation at the 3σ level. The ments in order to reach higher precision. The factor significance is still not enough to establish a break- 20 in precision obtained at BNL over the pioneer- down of the Standard Model in the muon g 2, i.e. a ing measurements performed at CERN has permit- − contribution from new physics. However it is a quan- ted to reach the electroweak scale in this process. titative and very valuable information which will help Another factor of 4, as anticipated by the new pro- to constrain the new physics, if it is found at the LHC. posal [49] submitted to Fermilab, or with the JPARC This is the present situation. It should and will project [50], will definitely provide quantitative in- evolve as new results and new initiatives are taking formation as we move into the new physics territory. place: 7) The hadronic spectral functions will continue to I would like to thank A. Höcker, B. Malaescu, be refined as new results are expected from BABAR G. López Castro, X.H. Mo, G. Toledo Sánchez, in the multihadronic channels, from KLOE (ππ) with P. Wang, C.Z. Yuan, and Z. Zhang for our fruitful improved approaches (preliminary results are already collaboration, and C.Z. Yuan and his colleagues for available for the large-angle ISR analysis [32]), and running a perfect workshop. from the upgraded CMD-3 and SND detectors at the

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