PHYSICAL REVIEW D 97, 114025 (2018)
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Muon − 2andαð 2 Þ: A new data-based analysis g MZ † ‡ Alexander Keshavarzi,1,* Daisuke Nomura,2,3, and Thomas Teubner1, 1Department of Mathematical Sciences, University of Liverpool, Liverpool L69 3BX, United Kingdom 2KEK Theory Center, Tsukuba, Ibaraki 305-0801, Japan 3Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan
(Received 6 April 2018; published 25 June 2018)
This work presents a complete reevaluation of the hadronic vacuum polarization contributions to the had;VP anomalous magnetic moment of the muon, aμ , and the hadronic contributions to the effective QED Δα ð 2 Þ þ − → coupling at the mass of the Z boson, had MZ , from the combination of e e hadrons cross section data. Focus has been placed on the development of a new data combination method, which fully incorporates all correlated statistical and systematic uncertainties in a bias free approach. All available þ − → e e hadrons cross section data have been analyzed and included, where the new datap compilationffiffiffi has yielded the full hadronic R-ratio and its covariance matrix in the energy range mπ ≤ s ≤ 11.2 GeV. Using these combined data and perturbative QCD above that range results in estimates of the hadronic had;LO VP −10 vacuum polarization contributions to g − 2 of the muon of aμ ¼ð693.26 2.46Þ × 10 and had;NLO VP −10 aμ ¼ð−9.82 0.04Þ × 10 . The new estimate for the Standard Model prediction is found to be SM −10 aμ ¼ð11659182.04 3.56Þ × 10 , which is 3.7σ below the current experimental measurement. The prediction for the five-flavor hadronic contribution to the QED coupling at the Z boson mass is Δαð5Þ ð 2 Þ¼ð276 11 1 11Þ 10−4 α−1ð 2 Þ¼128 946 0 015 had MZ . . × , resulting in MZ . . . Detailed comparisons with results from similar related works are given.
DOI: 10.1103/PhysRevD.97.114025
I. INTRODUCTION prediction is also improved to determine whether the − 2 ¼ g discrepancy is well established. The anomalous magnetic moment of the muon, aμ SM ð − 2Þ 2 The uncertainty of aμ is completely dominated by the g μ= , stands as an enduring test of the Standard Model had hadronic contributions, aμ , attributed to the contributions (SM), where the ∼3.5σ (or higher) discrepancy between the exp SM from the nonperturbative, low energy region of hadronic experimental measurement aμ and the SM prediction aμ resonances. The hadronic contributions are divided into the could be an indication of the existence of new physics exp hadronic vacuum polarization (VP) and hadronic light-by- beyond the SM. For aμ , the value is dominated by the light (LbL) contributions, which are summed to give measurements made at the Brookhaven National Laboratory – (BNL) [1 3], resulting in a world average of [4] had had;VP had;LbL aμ ¼ aμ þ aμ : ð1:2Þ
exp −10 aμ ¼ 11659209.1ð5.4Þð3.3Þ × 10 : ð1:1Þ This analysis, KNT18, is a complete reevaluation, in line with previous works [7–9], of the hadronic vacuum had;VP Efforts to improve the experimental estimate at Fermilab polarization contributions, aμ . The hadronic vacuum (FNAL) [5] and at J-PARC [6] aim to reduce the exper- polarization contributions can be separated into the leading- imental uncertainty by a factor of 4 compared to the BNL order (LO) and higher-order contributions, where the LO measurement. It is therefore imperative that the SM and next-to-leading order (NLO) contributions are calcu- lated in this work.1 These are calculated utilizing dispersion *[email protected] † integrals and the experimentally measured cross section, [email protected] ‡ [email protected] σ0 ð Þ ≡ σ0ð þ − → γ → þ γÞ ð Þ had;γ s e e hadrons ; 1:3 Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to 1The next-to-next-to-leading order (NNLO) contributions have SM the author(s) and the published article’s title, journal citation, recently been determined in [10] and are included as part of aμ and DOI. Funded by SCOAP3. below.
2470-0010=2018=97(11)=114025(27) 114025-1 Published by the American Physical Society KESHAVARZI, NOMURA, and TEUBNER PHYS. REV. D 97, 114025 (2018) where the superscript 0 denotes the bare cross section The structure of this paper is as follows. Section II (undressed of all vacuum polarization effects) and the describes all changes to the treatment and combination of subscript γ indicates the inclusion of effects from final hadronic cross section data since [9]. Section III gives had;LO VP state photon radiation. At LO, the dispersion relation reads details concerning the contributions to aμ and Z Δα ðM2 Þ from the different hadronic final states and α2 ∞ had Z had;LO VP ds aμ ¼ RðsÞKðsÞ; ð1:4Þ energy regions, culminating in updated estimates of both 2 2 3π s s had;LO VP had;NLO VP Δα ð Þ th aμ , aμ , and had MZ . These results are 2 then compared with other works and the new prediction of where α ¼ αð0Þ, s ¼ mπ, RðsÞ is the hadronic R-ratio had;VP th aμ is combined with all other SM contributions to given by SM determine the SM prediction aμ , with details given σ0 ðsÞ σ0 ðsÞ concerning the consequences of this on the existing ð Þ¼ had;γ ≡ had;γ ð Þ g − 2 discrepancy and the new result for αðM2 Þ. Finally, R s σ ð Þ 4πα2 ð3 Þ ; 1:5 Z pt s = s the conclusions of this work, along with discussions of the prospects for g − 2 in the future and potential improve- and KðsÞ is a well known kernel function [11,12],given SM ments for aμ , are presented in Sec. IV. also in Eq. (45) of [7], but differing by a normalization 2 factor of mμ=ð3sÞ. This kernel function [which behaves as 2 II. CHANGES SINCE THE LAST ANALYSIS KðsÞ ∼ mμ=ð3sÞ at low energies], coupled with the factor of (HLMNT11) 1=s in the integrand of Eq. (1.4), causes the hadronic vacuum polarization contributions to be dominated by the A. Radiative corrections low energy domain. At NLO, the data input is identical, with corresponding dispersion integrals and kernel func- 1. Vacuum polarization corrections tions [13] (see also [7]). In the previous analysis [9] Equations (1.4) and (1.8) require the experimental cross (denoted as HLMNT11), the LO hadronic vacuum polari- section to be undressed of all VP effects as VP (running zation contributions were found to be coupling) corrections to the hadronic cross section are had;VP counted as part of the higher order contributions to aμ . had;LO VP −10 aμ ðHLMNT11Þ¼ð694.91 4.27Þ × 10 ; ð1:6Þ Any new and old data that have not been corrected for VP effects require undressing. However, recent data are more which resulted in the total SM prediction of commonly undressed in the experimental analyses already, removing the need to apply a correction to these data sets. SMð 11Þ¼ð11659182 8 4 9Þ 10−10 ð Þ aμ HLMNT . . × : 1:7 This benefits the data combination as new, more precise data undressed of VP effects are dominating the fit for This, compared with the experimental measurement, gave a many channels which, in turn, reduces the impact of the g − 2 discrepancy of 3.3σ. had;VP extra radiative correction uncertainty which is applied to In addition to calculating aμ , the combination of each channel. hadronic cross section data is also used to calculate the Following the previous analyses, the “bare” cross section αð 2Þ 0 running (momentum dependent) QED coupling, q . σ in Eq. (1.5) is determined to be This, in particular, is then used to determine the effective had QED coupling at the Z boson mass, αðM2 Þ, which is the σ0 ¼ σ ð Þ Z had Cvp had; 2:1 least precisely known of the three fundamental electroweak (EW) parameters of the SM [the Fermi constant GF, MZ, where C is a multiplicative VP correction. For this αð 2 Þ vp and MZ ] and hinders the accuracy of EW precision fits. purpose, the self-consistent KNT18 vacuum polarization had;VP Using an identical data input as that used for aμ , the routine, vp_knt_v3_0, has been updated via an iteration hadronic contributions to the effective QED coupling are of the new data input.2 An in-depth discussion of the determined from the dispersion relation determination of the vacuum polarization and the corre- Z sponding routine will be given in [14]. 2 ∞ ð5Þ 2 αq RðsÞ The decisions to undress all data sets previously cor- Δα ðq Þ¼− P ds ; ð1:8Þ had 3π ð − 2Þ rected in [9] are unchanged, with the exception of two sth s s q measurements of the inclusive hadronic R-ratio by the where the superscript (5) indicates the contributions Crystal Ball Collaboration [15,16]. A reexamination of the from all quark flavors except the top quark, which is experimental analyses of these two measurements has added separately. Together with the leptonic contributions, 2 αð Þ¼ 2 this is used to determine the running coupling q This routine is available for use by contacting the authors α ð1 − Δα ð 2Þ − Δα ð 2ÞÞ = had q lep q . directly.
114025-2 − 2 αð 2 Þ 97, MUON g AND MZ : A NEW DATA-BASED ANALYSIS PHYS. REV. D 114025 (2018) shown that not only do they include some treatment of the ð0Þ ð0Þ α σ ðsÞ¼σ ðsÞ 1 þ ηðsÞ ; ð2:4Þ VP, but that they also both account for this correction with had;ðγÞ had π sizable systematic uncertainties. Applying an extra VP correction here would be an overestimate of the correction has been used.3 Here, the function ηðsÞ is given e.g. in [22] and its corresponding uncertainty. Therefore, undressing of and the subscript γ indicates the one photon inclusive cross these data is not applied here. In addition, it should be noted section. However, experimental cross section measure- that in a separate work [17], two measurements by the ments, by nature, include all virtual and soft real radiation KLOE Collaboration [18,19] in the πþπ− channel are now effects. Therefore, to estimate possibly missing photon undressed of VP effects using an updated routine [20] FSR, ideally only the effects from (hard) real radiation compared to the one used previously [21]. above/within resolution/cut parameters which are specific The undressing of narrow resonances in the cc¯ and bb¯ for a given experiment or analysis have to be estimated. regions requires special attention. Importantly, the elec- Whereas in the calculation of the inclusive correction a tronic width of an individual resonance, Γee, must be regularization of the virtual and real soft contributions is undressed of vacuum polarization effects using a para- required to obtain the infrared finite result η, the hard real metrization of the VP where the correction excludes the radiation, ηhard;real, can be estimated numerically from contribution of that resonance, such that Z pffiffi s−2 sΛ ηhard;realð Þ¼ 0ρ ð 0Þ ð Þ 2 2 s ds fin s; s ; 2:5 ðα=α ðM ÞÞ 4 2 Γ0 ¼ no res res Γ : ð2:2Þ m ee 1 þ 3α=ð4πÞ ee where m is the mass of the (scalar) particle, Λ is a finite α infrared cutoff parameter on the invariant mass of the Here, no res is the effective QED coupling neglecting the ρ contribution of the resonance itself and is given by emitted photon, and fin is the radiator function (see Appendix B of [23]). α In the case of the KþK− channel, by far the largest α ðsÞ ≡ ; ð : Þ no res 1 − Δα ð Þ 2 3 contribution to aμ and Δα (and their errors) comes from the no res s energy region of the ϕ peak, where the phase space for real Δα ð Þ radiation is severely restricted. While the calculation of real where no res s is determined from Eq. (1.8) such that the input RðsÞ does not include the resonance that is being radiation accounting for all experimental cuts would be very corrected. To include the resonance would lead to an complicated and beyond the scope of this work, an estimate Λ inconsistent definition of the narrow resonance. can easily be made based on Eq. (2.5), by relating to the cuts in the photon accolinearity given in individual exper- imental analyses. In Fig. 1, the result for the fully inclusive 2. Final state radiative corrections correction ηðsÞ (left panel) is compared to the estimates for Final state radiation (FSR) cannot be separated in an the real hard radiation, where ηhard;realðsÞ (right panel) now unambiguous way in the measured hadronic cross sections. depends on the accolinearity cuts as given by the two Therefore, while formally of higher order in α, FSR experimental analyses depicted. Clearly, at and around photons have to be taken into account in the definition the ϕ peak, phase space restrictions strongly suppress any of the one particle irreducible hadronic blobs and will hard real radiation and using the inclusive correction would already be included as part of the leading order hadronic lead to an overestimate of the possible effects. Given the VP contributions. However, depending on the experimental small size of both the possible correction in the ϕ region and þ − analyses, some amount of real photon FSR may have been the contribution of the K K channel above the ϕ to both removed during the event selection. Adding back these mean value and error of aμ and Δα, no correction or missed contributions is model dependent and not feasible additional error estimate due to FSR is now applied in the þ − for general hadronic final states. It is therefore necessary to K K channel. For the neutral kaon channel, hard photon estimate the possible effects and their impact on the radiation (which would resolve the quark charges) is accuracy of the data compilations. Similar to the case of similarly suppressed and no FSR correction or additional VP corrections, an extra radiative correction uncertainty error is applied in this channel also. due to FSR effects is then estimated channel by channel. The situation is different in the two pion channel. A For the important πþπ− and KþK− channels, detailed study similar to the two kaon channel showed that in studies have been performed for this analysis. Here, and principle larger contributions from real radiation of the especially in the limited energy range below 2 GeV, it has order of the inclusive correction can arise. However, these been shown that scalar QED provides a good description of contributions are strongly dependent on the cut applied in photon FSR; see e.g. [22–24]. In the past, to estimate þ − þ − possible FSR effects in π π and K K production, the 3Here, the term “fully inclusive” means inclusive of effects fully inclusive, order α correction to the cross section, from virtual and real (both soft and hard) one-photon emission.
114025-3 KESHAVARZI, NOMURA, and TEUBNER PHYS. REV. D 97, 114025 (2018)
1.00025 K+K- CFSR CMD-2 (95) 1.05 2000 2000 σ0(e+e- → K+K-) SND (00) 1.0002 σ0(e+e- → K+K-) π 1500 / 1500
1.045 α ) [nb] ) [nb] - - π / K (s)
1.00015 K α + + K K (s) η → →
1000 1000 -
1.04 - hard,real e e 1.0001 η + 1 + + (e (e 0 0 1 + σ σ 500 500 1.035 1.00005
1.03 0 1 0 1.01 1.015 1.02 1.025 1.03 1.035 1.04 1.045 1.05 1.01 1.015 1.02 1.025 1.03 1.035 1.04 1.045 1.05 √s [GeV] √s [GeV]
FIG. 1. The effect of final state radiation in the KþK− channel in the ϕ resonance region. Left panel: the fully inclusive FSR correction ηðsÞ. Right panel: hard real radiation ηhard;realðsÞ, estimated with accolinearity cuts used in the two analyses [25,26]. The eþe− → KþK− cross section is also plotted for reference.
1400 experimental analysis, they are (at least to some extent) π+π- 1.012 CFSR estimated by Monte Carlo (MC) simulation and contribute to σ0(e+e- → π+π-) 1200 the systematic uncertainties.4 However, for many data sets, it 1.01 is far from clear to which extent FSR effects are included in 1000 the systematic errors. Therefore, possible effects are ) [nb] π 1.008 - / π
α accounted for by applying an additional uncertainty deter- 800 + π
(s)
η mined as a fraction of the respective contribution. →
1.006 600 - e 1 + + (e 1.004 0 3. Estimating extra uncertainty due to 400 σ radiative corrections 1.002 200 As in [7–9], extra uncertainties are estimated whenever
1 0 additional radiative corrections are applied. This is done, 0.4 0.6 0.8 1 1.2 1.4 first and foremost, to account for any under- or over- √s [GeV] correction that may occur due to a lack of information ηð Þ concerning the treatment of radiative corrections in the FIG. 2. The behavior of the inclusive FSR correction, s , for experimental analyses. However, these radiative correction the process eþe− → πþπ−. The eþe− → πþπ− cross section is also plotted for reference. uncertainties also account for any possibly incorrect treat- ment in the analyses, e.g. missed FSR or inconsistent subtraction of VP contributions. This is especially true for Eq. (2.5) and would require a more detailed, measurement- older data, where there is very little or even no information by-measurement analysis, which is beyond the scope of this at all regarding how the data have been treated. work. In addition, for many of the data sets the required vp In each channel, the difference Δaμ between the detailed information is not available. Therefore, in sets estimates of aμ with and without additional VP corrections which are understood to not include the full FSR correc- is determined. For the uncertainty due to VP, one-third of tions, the fully inclusive correction (as shown in Fig. 2)is þ − the shift applied to the respective measurements. In the π π data combination, recent sets from radiative return, where 1 δ vp ¼ Δ vp ð Þ additional photons are part of the leading order cross aμ 3 aμ 2:6 section and are studied in detail as part of the analyses, have now become dominant. Therefore, the impact of the is taken. This is reduced from one-half in [9], safe in the fully inclusive FSR correction to older sets is suppressed knowledge that the KNT18 VP routine [14], which is for both mean value and error in comparison to previous determined iteratively in a self-constistent way, is accurate analyses. The procedure to determine the corresponding to the level of a permille when correcting the cross section error estimate is discussed in the next section, while resulting numbers are given in Sec. III A, which contains 4In the recent work [27], a version of the MC program the detailed analysis of the πþπ− channel. CARLOMAT [28] is discussed, which includes real photon radi- For the subleading, multihadron channels, there are, at ation in multihadron channels. This could be used to further study present, no equivalent FSR calculations. Depending on the the FSR effects.
114025-4 − 2 αð 2 Þ 97, MUON g AND MZ : A NEW DATA-BASED ANALYSIS PHYS. REV. D 114025 (2018) and that newer data sets are commonly undressed of VP 2. Fitting data effects by the experiment with a modern routine. The previous analyses [7–9] employed a nonlinear χ2- For the extra uncertainties due to FSR, there are now no þ − 0 0 minimization utilizing fitted renormalization factors as contributions from the K K and KSKL channels (see the nuisance parameters that represented the energy indepen- πþπ− discussion above). For the channel, the full difference dent systematic uncertainties. Although a powerful method, πþπ− between the estimates of aμ with and without additional recent literature [30] (see also [31]) has highlighted the FSR corrections is taken as the FSR uncertainty. For all possibility that an incorrect treatment of multiplicative other channels, including the inclusive data combination normalization uncertainties in a χ2 minimization can incur above 1.937 GeV, a fraction of 1% of the respective cross a systematic bias (see Chap. 4 of [30]). In addition, section is applied as the uncertainty. although the nonlinear χ2-minimization used in [9] was The numerical estimates of all additional radiative adjusted to include covariance matrices, the method’s correction uncertainties are given in the respective sections reliance on fitting energy independent renormalization for the individual channels. The same procedures are factors prevented the use of correlated uncertainties to Δα applied in the calculation of the contributions to had. their full capacity. As recent precise data, specifically radiative return measurements in the πþπ− and KþK− B. Data combination channels, have been released with energy dependent uncertainties and nontrivial bin-to-bin correlations for both The combination of σ0ðeþe− → hadronsÞ data and its the statistical and the systematic uncertainties, the fit corresponding uncertainty has undergone a great deal of procedure had to be modified to allow the full use of all scrutiny since [9]. This has been due to the ever increasing available correlations in a bias-free approach. Therefore, amount of data that have become available and due to a instead of fitting renormalization factors (nuisance param- better understanding of the treatment of correlated uncer- eters), an iterative fit procedure as advocated in [30] has tainties that are now a prominent feature of many cross been adopted, which reinitializes the full covariance matri- section measurements. In [9], covariance matrices in the πþπ− ces at each iteration step and, consequently, avoids bias. channel from BABAR [29] and KLOE [18,19] for Previously, in [7–9], a constant cross section had been statistical and systematic uncertainties had already been assumed across the width of each cluster. In this work, included based on the method of fitted renormalization the fitted cross section values at the cluster centers are factors. In the following, a new method for the combination obtained from the iterative χ2-minimization where the cross σ0 of had data is introduced, which will allow one to fully take section is taken to be linear between adjacent cluster into account all correlated uncertainties. centers. This allows for a more stable fit and is consistent had;VP with the trapezoidal rule integration utilized for the aμ pffiffiffi 1. Clustering data points Δα ¼ ðmÞ and had integrals. If data points at energies s Ei Within each hadronic channel, data points from different are combined into cluster m, then the weighted average of 5 experiments are assigned to clusters (see [7–9] for details). the cross section value Rm and energy Em for the cluster In this work, the clustering algorithm has been improved. It center are is universal for all different channels and only differs in the " #" # ð Þ ð Þ −1 δ δ XN m ðmÞ XN m assigned maximum cluster size (and res, a maximum R 1 R ¼ i ð2:7Þ cluster size applicable at individual narrow resonances). m ð ˜ ðmÞÞ2 ð ˜ ðmÞÞ2 δ δ i¼1 dRi i¼1 dRi A scan over (and res if applicable) is performed to determine a suitable clustering configuration which must and avoid both over- and underclustering. Too wide or over- " #" # populated clusters would effectively lead to a rebinning of ðmÞ ð Þ ðmÞ −1 XN E m XN 1 data points from individual experiments and risk loss of E ¼ i ; ð2:8Þ information, while a too narrow clustering would result, in m ð ˜ ðmÞÞ2 ð ˜ ðmÞÞ2 i¼1 dRi i¼1 dRi the extreme, in an erratic point-to-point representation of the cross section and no gain in the accuracy. The preferred ðmÞ 2 where Ri is the cross section value of data point i χ = : : : ð Þ configuration is then chosen based on the global min d o f contributing to cluster m, N m is the total number of data had;VP and the uncertainty on aμ , combined with checks by points contributing to cluster m, and eye of the resulting spectral function. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ˜ ðmÞ ¼ ð ðmÞ Þ2 þð ðmÞ Þ2 ð Þ dRi dRi;stat dRi;sys : 2:9 5 χ2 The effect that different clusterings have on the min=d:o:f. of ðmÞ ðmÞ had;VP the fit, the resulting aμ value, and its error has been discussed dRi;stat and dRi;sys denote the absolute statistical and in detail in [9]. systematic uncertainties, respectively. With a linear cross
114025-5 KESHAVARZI, NOMURA, and TEUBNER PHYS. REV. D 97, 114025 (2018) section now assumed, if data point i belongs to cluster m where ð Þ and E m >E , then its interpolant cross section value Ri i m m ðmÞ ðnÞ is given by XN XN −1ð ðiÞ ðjÞÞ¼ C−1ð ðmÞ ðnÞÞ ð Þ VI m ;n I i ;j : 2:15 ð Þ i¼1 j¼1 ðE m − E Þ Riþ ¼ þ i m ð − Þ ð Þ m Rm ð − Þ Rmþ1 Rm ; 2:10 Emþ1 Em The solution to this yields the cluster centers Rm and the covariance matrix VIðm; nÞ which describes the correlation þ ðmÞ where the indicates that Ei >Em. If, on the other hand, between the errors dRm and dRn. As in Eqs. (2.10) and ðmÞ Rj Rjþ ðnÞ Ei
114025-6 − 2 αð 2 Þ 97, MUON g AND MZ : A NEW DATA-BASED ANALYSIS PHYS. REV. D 114025 (2018)
1400 Linear Linear 2000 1200 Quadratic Quadratic Cubic Cubic 1000 1500 ) [nb] ) [nb] − − π Κ + + 800 π Κ
→ →
1000 −
600 − e e + + (e (e 0 400 0 σ σ 500 200
0 0 0.6 0.65 0.7 0.75 0.8 0.85 0.9 1.01 1.015 1.02 1.025 1.03 √s [GeV] √s [GeV] (a) (b)
FIG. 3. The differences observed using linear, quadratic, and cubic integration routines in prominent resonance regions in the πþπ− and KþK− channels.
had;LO VP ¼ had;LO VP ðδ had;LO VPÞ ðδ had;LO VPÞ resonances if data are sparse. However, with the current aμ aμ aμ stat aμ sys density of cross section measurements, especially in the ðδ had;LO VPÞ ðδ had;LO VPÞ dominant hadronic channels, the differences between trap- aμ vp aμ fsr ezoidal rule integration and any higher order polynomial ¼ had;LO VP ðδ had;LO VPÞ ð Þ aμ aμ tot: 3:1 approximation are consequently small and of no concern. −10 This can be seen in the plots in Fig. 3. All results for aμ are given in units 10 . For the had;VP Δα The calculation of the uncertainty of aμ and had Δα ð 2 Þ contributions to had MZ , only the mean value with has been modified to improve the determination of the error the corresponding total uncertainty, contribution at the integral boundaries. Should the upper or lower integral boundary, Ea, with energy Em 114025-7 KESHAVARZI, NOMURA, and TEUBNER PHYS. REV. D 97, 114025 (2018) πþπ− combination. Interestingly, however, it is in better Fit of all π+π− data: 369.41 ± 1.32 agreement with the BABAR data at the peak of the resonance where the cross section is largest. Although Direct scan only: 370.77 ± 2.61 BABAR still influences with an increase, the agreement between the other radiative return measurements and the KLOE combination: 366.88 ± 2.15 direct scan data largely compensates for this effect. This is demonstrated by Fig. 6, with the combination clearly BaBar (09): 376.71 ± 2.72 favoring the other measurements. Tension between data sets, however, still exists and is reflected in the local χ2 BESIII (15): 368.15 ± 4.22 error inflation, which results in an ∼15% increase in the πþπ− uncertainty of aμ . The effect of this energy dependent 360 365 370 375 380 385 390 395 error inflation is shown in Fig. 7, where the difference in π+π− 10 aμ (0.6 ≤ √⎯s ≤ 0.9 GeV) x 10 using a local scaling of the error instead of a global one is clearly visible. Tensions arise in particular in the ρ FIG. 4. The comparison of the integration of the individual resonance region, where the cross section is large. radiative return measurements and thep combinationffiffiffi of direct scan πþπ− þ − The full combination of all data is found to give π π measurements between 0.6 ≤ s ≤ 0.9 GeV. pffiffiffi πþπ− aμ ½0.305 ≤ s ≤ 1.937 GeV of the KLOE data, and a comparison with other exper- ¼ 502.97 1.14 1.59 0.06 0.14 imental measurements of σππðγÞ are presented. These ¼ 502.97 1.97 ð3:3Þ covariance matrices are used here as input in the full þ − π π combination in order to fully incorporate the corre- and lation information of the KLOE data as an influence on pffiffiffi þ − 2 π π Δα þ − ð Þ½0 305 ≤ ≤ 1 937 both the estimate of aμ and its uncertainty. π π MZ . s . GeV The BESIII measurement in the ρ resonance region ¼ 34.26 0.12: ð3:4Þ (again with full statistical and systematic covariance πþπ− matrices) allows for an in-depth comparison of the existing Although this value of aμ stays well within the error radiative return measurements already contributing to the estimate of [9], it exhibits a substantial decrease of the πþπ− channel, namely the three measurements by the mean value. This has been attributed to the new data KLOE Collaboration and the finely binned measurement combination routine which allows for the full use of from the BABAR Collaboration [29].In[9], details were correlations in the determination of the mean value as well given regarding tension between the KLOE and BABAR as the uncertainty and the inclusion of the new, precise measurements, where the BABAR data were considerably radiative return data which suppresses the influence of higher. As is evident from Fig. 4, tension exists between the BABAR in the ρ resonance region. BABAR data and all other contributing data in the dominant In comparison with Eq. (3.3), the BABAR data alone in the πþπ− ρ region. When considering this along with the plots of the same energy range give an estimate of aμ ðBABARÞ¼ resulting cross section in Fig. 5, it is clear that the new 513.2 3.8. Should all available πþπ− data be combined BESIII data agrees well with the KLOE data and the full using a simple weighted average as in Eq. (2.7), which only 1400 1400 BESIII (15) BESIII (15) 1200 KLOE combination 1300 KLOE combination CMD-2 (06) CMD-2 (06) 1200 1000 SND (04) SND (04) ) [nb] ) [nb] - CMD-2 (03) - 1100 CMD-2 (03) π π + 800 + π+π- π+π- π Fit of all data π Fit of all data 1000 → BaBar (09) → BaBar (09) - 600 - e e + + 900 (e (e 0 0 σ 400 σ 800 200 700 0 600 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 0.74 0.75 0.76 0.77 0.78 0.79 0.8 0.81 0.82 √s [GeV] √s [GeV] FIG. 5. Contributing data in the ρ resonance region of the πþπ− channel plotted against the new fit of all data (left panel), with an enlargement of the ρ − ω interference region (right panel). 114025-8 − 2 αð 2 Þ 97, MUON g AND MZ : A NEW DATA-BASED ANALYSIS PHYS. REV. D 114025 (2018) 0.4 π+π- + - 0 BESIII (15) a (0.6 ≤ ⎯√s ≤ 0.9 GeV) = (369.41 ± 1.32) x 10-10 Fit of all π π π data μ 1400 1000 KLOE combination √ χ2 SND (15) Global ( min/d.o.f.) = 1.30 CMD-2 (06) CMD-2 (07) Scans BaBar (04) 0.3 SND (04) 1200 SND (02,03) CMD-2 (03) 100 CMD-2 (95,98,00) π+π- Fit of all data DM2 (92) BaBar (09) 1000 ) [nb] ND (91) 0.2 ) [nb] - 0 Fit 0 + - + - σ (e e → π π ) 0 π σ CMD (89) π + )/ - 800 π DM1 (80) π 10 0 Fit + σ → π 0.1 - - 0 600 e σ + → ( - (e 0 e 1 + 0 400 σ (e 0 200 σ -0.1 0.1 0 0.6 0.65 0.7 0.75 0.8 0.85 0.9 √s [GeV] 0.01 0.8 1 1.2 1.4 1.6 1.8 √ FIG. 6. The relative difference of the radiative return and s [GeV] πþπ− important direct scan data sets contributing to aμ and the fit σ0ð þ − → πþπ−π0Þ of all data. For comparison, the individual sets have been FIG. 8.pffiffiffi The cross section e e in the range 0 66 ≤ ≤ 1 937 ω ϕ normalized against the fit and have been plotted in the ρ region. . s . GeV, where the prominent and reso- The light green band represents the BABAR data and their errors nances are clearly visible. (statistical and systematic, added in quadrature). The yellow band represents the full data combination which incorporates all the energy dependent correlations. In addition, the radiative correlated statistical and systematic uncertainties. However, the corrections uncertainties have reduced since [9], as dis- width of the yellow band simply displays the square root of the diagonal elements of the total output covariance matrix of the fit. cussed in detail in Sec. II A. B. π + π − π0 channel 1400 √ χ2 Local ( min/d.o.f.) Since [9], there has been only one new addition to the 3 √ χ2 Global ( min/d.o.f.) = 1.30 1200 πþπ−π0 σ0(e+e− → π+π−) channel [36]. This new data set improves this 2.5 1000 channel away from resonance, where previously only ) [nb] − BABAR data [37] had provided a contribution of notable 800 π + π /d.o.f.) precision. Compared to [9], an additional change is applied to 2 → min ϕ 2 three separate data scans over the resonance in a meas- 600 − χ e ( + √ urement by CMD-2 [38], where the systematic uncertainties (e 1.5 400 0 σ between the three scans are now taken to be fully correlated 200 [39]. These changes, along with the new data combination 1 routine, have resulted in an improved estimate of 0 pffiffiffi 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 πþπ−π0 μ ½0 66 ≤ ≤ 1 937 √s [GeV] a . s . GeV ¼ 47.79 0.22 0.71 0.13 0.48 FIG. 7. The effect of the local χ2 inflation and the overall global χ2 πþπ− ¼ 47.79 0.89 ð3:5Þ min=d:o:f: in the channel, which is plotted against the þ − þ − e e → π π cross section for reference. and pffiffiffi 2 Δα þ − 0 ðM Þ½0.66 ≤ s ≤ 1.937 GeV ¼4.77 0.08: provides the error weighting to each cluster by its local π π π Z πþπ− πþπ− ð Þ uncertainty, the estimate for aμ would be aμ ðnaive 3:6 Þ¼509 1 2 9 weighted average . . . In this case, the estimate Figure 8 shows the full integral range of the data for the is strongly pulled up by the fine binning and high statistics of þ − 0 π π π cross section. Figure 9 shows an enlargement of the the BABAR data that dominate when no correlations are ω and ϕ resonance regions in this channel. taken into account for the mean value. This difference of nearly 2σ when comparing to Eq. (3.3) indicates the C. 4π channels importance of fully incorporating all available correlated uncertainties in any combination of the data. The πþπ−πþπ− channel now includes two new additions The uncertainty has reduced by approximately one-third. since [9]. First, an improved statistics measurementpffiffiffi by Again, this is due to the new, precise radiative return data the BABAR Collaboration in the range 0.6125 ≤ s ≤ which further dominate the πþπ− fit and the improvement 4.4875 GeV [40] supersedes their previous measurement of the overall combination which now fully incorporates in this channel [41]. More recently, a data set by the CMD-3 114025-9 KESHAVARZI, NOMURA, and TEUBNER PHYS. REV. D 97, 114025 (2018) 1600 + - 0 + - 0 Fit of all π π π data 700 Fit of all π π π data SND (15) SND (15) 1400 CMD-2 (07) Scans 600 CMD-2 (07) Scans 1200 BaBar (04) BaBar (04) SND (02,03) SND (02,03) ) [nb] ) [nb] 500 0 CMD-2 (95,98,00) 0 CMD-2 (95,98,00) π 1000 π - - π DM2 (92) π DM2 (92) + + 400 π π 800 ND (91) ND (91) → → CMD (89) CMD (89) - - 300 e e + 600 DM1 (80) + DM1 (80) (e (e 0 0 200 σ 400 σ 200 100 0 0 0.75 0.76 0.77 0.78 0.79 0.8 0.81 0.82 1 1.005 1.01 1.015 1.02 1.025 1.03 1.035 1.04 √s [GeV] √s [GeV] (a) (b) FIG. 9. Enlargements of the resonance regions in the πþπ−π0 final state. pffiffiffi Collaboration in the range 0.92 ≤ s ≤ 1.06 GeV [42] has The picture for the πþπ−π0π0 final state has also improved, been completed, which better resolves the interference with a new measurement of this channel by BABAR [44] pattern of the ϕ → πþπ−πþπ− transition that is clearly providing the only new data in this channel since 2003. As evident in the nonresonant cross section. In addition, the with the πþπ−πþπ− channel, the M3N data [43] in this M3N thesis data [43] that are not published, but were channel have been omitted. The estimate for this channel is pffiffiffi previously included in this channel, are now discarded. πþπ−π0π0 With these changes, aμ ½0.850 ≤ s ≤ 1.937 GeV pffiffiffi πþπ−πþπ− ¼ 19.39 0.09 0.74 0.04 0.19 aμ ½0.6125 ≤ s ≤ 1.937 GeV ¼ 19.39 0.78 ð3:9Þ ¼ 14.87 0.02 0.13 0.03 0.15 and ¼ 14.87 0.20 ð3:7Þ pffiffiffi 2 Δαπþπ−π0π0 ðM Þ½0.850 ≤ s ≤ 1.937 GeV ¼5.00 0.20; and Z pffiffiffi ð3:10Þ 2 Δα þ − þ − ð Þ½0 6125 ≤ ≤ 1 937 π π π π MZ . s . GeV where there is clear improvement since [9]. The uncertainty ¼ 4.02 0.05: ð3:8Þ þ − 0 0 contribution from π π π π is, however, still relatively large had;LO VP Δα Here, the mean value has increased since [9] largely due to in comparison with its contribution to aμ and had the new BABAR data. The fitted cross section and data are and requires better new data. The new fit of the data for the displayed in Fig. 10. bare cross section eþe− → πþπ−π0π0 is shown in Fig. 11. 50 60 BaBar (17) Fit of all π+π-π+π- data SND (03) CMD-3 (16) CMD-2 (99) BaBar (12) 50 DM2 (90) 40 CMD-2 (04) OLYA (86) SND (03) MEA (81) ) [nb] ) [nb] - CMD-2 (00,00) 0 π 40 GG2 (80) π + ND (91) 30 0 + - 0 0 π π π π π π Fit of all data - DM2 (90) - π π Fit w/o BaBar data + CMD (88) + π π 30 OLYA (88) → → DM1 (79,82) - 20 - e GG2 (81) e + + ORSAY (76) 20 (e (e 0 0 σ 10 σ 10 0 0 0.8 1 1.2 1.4 1.6 1.8 1 1.2 1.4 1.6 1.8 √s [GeV] √s [GeV] 0 þ − þ − þ − 0 þ − þ − 0 0 FIG. 10.p Theffiffiffi cross section σ ðe e → π π π π Þ in the range FIG. 11.pffiffiffi The cross section σ ðe e → π π π π Þ in the range 0.6125 ≤ s ≤ 1.937 GeV. 0.850 ≤ s ≤ 1.937 GeV. 114025-10 − 2 αð 2 Þ 97, MUON g AND MZ : A NEW DATA-BASED ANALYSIS PHYS. REV. D 114025 (2018) The fit without the new BABAR data is shown for comparison CMD-3 (17) SND (16) Scans to highlight the large improvement this new data set provides. 1000 Babar (13) SND (07) SND (00) Scans D. ¯ channels CMD-2 (95) KK CMD (91) ) [nb] 100 þ − - OLYA (81) The K K channel now includes a precise and finely K DM2 (87) + BCF (86) binned measurement by the BABAR Collaboration, sup- K DM2 (83) → DM1 (81) plemented with full statistical and systematic covariance - 10 e Fit of all K+K- data + matrices [45]. This is the first and only example to date of (e 0 σ the release of energy dependent, correlated uncertainties 1 outside of the πþπ− channel and has an overwhelming influence on the data combination. There is also a new measurement in this channel of the ϕ resonance by the 0.1 CMD-3 Collaboration [46]. The existing CMD-2 scans in 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 √ the same region [47] are omitted from this work as they s [GeV] suffer from an overestimation of the trigger efficiency for (a) slow kaons [46] and are awaiting reanalysis [48].In 2500 addition, two new scans by the SND Collaboration mea- CMD-3 (17) sured at the tail of the ϕ and into the continuum are SND (16) Scans Babar (13) included [49]. The systematic uncertainties of these two 2000 SND (07) SND (00) Scans scans, along with the existing two scans by SND [26], are CMD-2 (95) CMD (91) considered to be fully correlated [39]. The combination of ) [nb] - OLYA (81) K 1500 þ − DM2 (87) the available K K data now gives + BCF (86) K DM2 (83) þ − pffiffiffi → K K ½0 9875 ≤ ≤ 1 937 - DM1 (81) aμ . s . GeV e 1000 + - + Fit of all K K data (e ¼ 23 03 0 08 0 20 0 03 0 00 0 . . . . . σ 500 ¼ 23.03 0.22 ð3:11Þ and 0 pffiffiffi 1.01 1.015 1.02 1.025 1.03 2 √ Δα þ − ð Þ½0 9875 ≤ ≤ 1 937 ¼3 37 0 03 s [GeV] K K MZ . s . GeV . . ; (b) ð3:12Þ 0 þ − þ − FIG. 12.p Theffiffiffi cross section σ ðe e → K K Þ in the range which exhibits an increase of the mean value of more than 0.9875 ≤ s ≤ 1.937 GeV and an enlargement of the ϕ reso- 1σ from the estimate in [9] attributed to the inclusion of the nance. The large influence of the BABAR data (black squares) new BABAR and CMD-3 data. The cross section of the overwhelms the older data. process eþe− → KþK− is displayed in Fig. 12. The uncertainty has drastically improved since [9] with 0 0 K K pffiffiffi much of the change being due to a finer clustering over the aμ S L ½1.00371 ≤ s ≤ 1.937 GeV ϕ resonance after the inclusion of the new high statistics ¼ 13.04 0.05 0.16 0.10 0.00 BABAR data. Following from the discussion in Sec. II A 2, there is now no FSR correction applied to this channel and, ¼ 13.04 0.19 ð3:13Þ therefore, there is no extra radiative correction uncertainty due to FSR. It should also be noted that any FSR correction and KþK− would result in an increase of aμ , showing again the pffiffiffi 2 Δα 0 0 ð Þ½1 00371 ≤ ≤ 1 937 strong influence the new data have had in this channel to K K MZ . s . GeV increase the mean value since [9], where previously an FSR S L ¼ 1 77 0 03 ð Þ correction was applied. . . : 3:14 0 0 New data for the KSKL final state is included from the BABAR Collaboration above the ϕ resonance [50] and from Again, there is also no additional FSR correction uncer- the CMD-3 Collaboration on the ϕ [51]. Two existing tainty applied to this channel, with the reasoning in Sec. II measurements in this channel [26,52] have multiple data A2 enforced by the probability of photon emission being scans of which the systematic uncertainties are now taken highly suppressed for a neutral final state and given the to be fully correlated [39]. This combination results in a limited phase space. The cross section of the process þ − → 0 0 contribution of e e KSKL is displayed in Fig. 13. 114025-11 KESHAVARZI, NOMURA, and TEUBNER PHYS. REV. D 97, 114025 (2018) 0 0 14 1000 Fit of all K SK L data σ0(KKπ) [HLMNT(11) isospin estimate] CMD-3 (16) Scans 0 BaBar (14) 12 σ (KKπ) [Data] SND (06) 100 CMD-2 (03) 10 ) [nb] L 0 SND (00) - Charged Modes K S SND (00) - Neutral Modes 8 0 K CMD (95) Scans 10 [nb] 0 → DM1 (81) σ - 6 e + (e 0 4 σ 1 2 0.1 0 1 1.2 1.4 1.6 1.8 1.3 1.4 1.5 1.6 1.7 1.8 1.9 √ s [GeV] √s [GeV] (a) FIG. 14. The measured cross section σ0ðKKπÞ compared to the estimate from the previously used isospin relation. 0 0 Fit of all K SK L data 1400 CMD-3 (16) Scans BaBar (14) and 1200 SND (06) CMD-2 (03) pffiffiffi ) [nb] 2 L 1000 Δα πðM Þ½1.260 ≤ s ≤ 1.937 GeV ¼0.89 0.04: 0 SND (00) - Charged Modes KK Z K S SND (00) - Neutral Modes 0 800 ð3:17Þ K CMD (95) Scans → DM1 (81) - 600 e + In [9], the isospin estimate in the same energy range yielded (e 0 400 σ KKπð 11 Þ¼2 65 0 14 ð Þ 200 aμ HLMNT isospin estimate . . : 3:18 0 1.01 1.015 1.02 1.025 1.03 This good agreement between the HLMNT11 isospin √s [GeV] estimate and the data-based approach in this analysis is also (b) demonstrated in Fig. 14. For KK2π, BABAR have measured the previously miss- σ0ð þ − → 0 0 Þ 0 0 πþπ− 0 0πþπ− 0 0 2π0 FIG. 13. The cross section e e KSKL with an enlarge- ing modes KSKL , KSKS [50], KSKL [53], ment of the ϕ resonance. 0 π∓π0 and KSK [55], such that this contribution is now determined using E. KK¯ π and KK¯ 2π channels σð 2πÞ¼σð þ −π0π0Þþσð þ −πþπ−Þ 0 0 π0 KK K K K K Since [9], the neutral final state KSKL has been þ σð 0 ¯0πþπ−Þþσð 0 0 2π0Þ measured by BABAR [53] and SND [54], completing all K K KSKL modes that contribute to the KKπ final state and removing 0 ∓ 0 0 ∓ 0 þ σðK K π π ÞþσðK K π π Þ the reliance on isospin for this channel (other than K0 ≅ K0 ). S L S L þ − 0 0 þ − þ − Therefore, the KKπ cross section is now calculated using ≃ σðK K π π ÞþσðK K π π Þ 0 ¯0 þ − 0 0 0 0 ∓ 0 ∓ þ σð π π Þþσð 2π Þ σð πÞ¼σð π Þþσð π Þ K K KSKL KK KSK KLK 0 ∓ 0 þ − 0 0 0 0 þ 2σð π π Þ ð Þ þ σð π Þþσð π Þ KSK : 3:19 K K KSKL ≃ 2σð 0 π∓Þþσð þ −π0Þþσð 0 0 π0Þ KSK K K KSKL ; 0 ≅ 0 Here, again, it is assumed that KS KL and, hence, ð3:15Þ σð 0 ¯0πþπ−Þ¼σð 0 0 πþπ−Þþσð 0 0πþπ−Þ resulting in a contribution of K K KSKL KSKS pffiffiffi þσð 0 0 πþπ−Þ KKπ KLKL aμ ½1.260 ≤ s ≤ 1.937 GeV ≃σðK0K0 πþπ−Þþ2σðK0K0πþπ−Þ: ð3:20Þ ¼ 2.71 0.05 0.11 0.01 0.01 S L S S ¼ 2 71 0 12 ð Þ . . 3:16 Therefore, the estimate in this channel is now found to be 114025-12 − 2 αð 2 Þ 97, MUON g AND MZ : A NEW DATA-BASED ANALYSIS PHYS. REV. D 114025 (2018) 18 Fit of all inclusive R data σ0(KKππ) [HLMNT(11) isospin estimate] 4.6 16 BaBar Rb data (09) σ0(KKππ) [Data] CLEO (07) 14 4.4 CLEO (98) CUSB (82) Υ 12 (5s)[Breit-Wigner] + Rudsc[pQCD] 4.2 Υ (6s)[Breit-Wigner] + Rudsc[pQCD] 10 Rudsc[pQCD] [nb] R(s) 4 0 σ 8 6 3.8 4 3.6 2 3.4 0 10.5 10.6 10.7 10.8 10.9 11 11.1 11.2 1.4 1.5 1.6 1.7 1.8 1.9 √s [GeV] √s [GeV] FIG. 16. The fit of inclusive R data in the region of the newly FIG. 15. The cross section σ0ðKKππÞ compared to the previous included BABAR Rb data, where the BABAR Rb is shown as light estimate using isospin relations. blue markers. The resonance structures of the ϒð5SÞ and ϒð6SÞ pffiffiffi states are clearly visible. KK2π aμ ½1.350 ≤ s ≤ 1.937 GeV sum of the resonance parametrizations which are clearly ¼ 1.93 0.03 0.07 0.01 0.01 very different from the bb¯ cross section as measured by ¼ 1.93 0.08 ð3:21Þ BABAR. Note that apart from the CLEO(98) data point [59] at 10.52 GeV, the CLEO(07) data point [60] at 10.538 GeV and and the CUSB data point at 11.09 GeV [61], there are no pffiffiffi other data in this bb¯ resonance region. Δα ð 2 Þ½1 350 ≤ ≤ 1 937 ¼0 75 0 03 KK2π MZ . s . GeV . . : The other new data are recent precise measurementspffiffiffi by 1 84 ≤ ≤ ð3:22Þ the KEDR Collaboration: one set between . s p3.05ffiffiffi GeV [62] and two scans in the energy range 3.12 ≤ Comparing Eq. (3.21) with the HLMNT11 isospin estimate s ≤ 3.72 GeV [63]. For the latter, the systematic uncer- in the same energy range of tainties are taken to be fully correlated p[39]ffiffiffi . The fit of the inclusive data in the range 1.937 ≤ s ≤ 3.80 GeV is KK2π aμ ðHLMNT11 isospin estimateÞ¼2.51 0.35 ð3:23Þ shown in Fig. 17, which demonstrates the good agreement between KEDR and pQCD. In [9]pffiffiffi, the decision was made and examining Fig. 15, it is evident that the isospin to use pQCD in the range 2.6 ≤ s ≤ 3.73 GeV, where the relations provided a poor estimate of this final state. quality of inclusive data was poor, with an error inflated Using the data, KK2π contributes a much smaller mean value with a greatly reduced uncertainty. 4.5 Fit of all inclusive R data KEDR (16) BaBar Rb data (09) F. Inclusive R-ratio data 4 BESII (09) CLEO (07) The combination of inclusive hadronic R-ratio data BES (06) pffiffiffi BES (02) between 1.937 ≤ s ≤ 11.2 GeV has three new data 3.5 BES (99) MD-1 (96) additions since [9]. The first of these are the precise Crystal Ball (88) pffiffiffi R(s) LENA (82) BABAR Rb data between 10.54 ≤ s ≤ 11.20 GeV [56], 3 pQCD which must be deconvoluted of the effects from initial state radiation (ISR) and must have the radiative tails of the 2.5 resonances from the ϒð1S − 4SÞ states removed (see [57]). These data are then added to the perturbative QCD (pQCD) 2 estimate of Rudsc [58] to be included as an accurate data set 2 2.5 3 3.5 4 4.5 in the inclusive channel in this region. The inclusion of the √s [GeV] BABAR Rb data is particularly beneficial as it resolves the resonances of the ϒð5SÞ and ϒð6SÞ states, removing FIG. 17.pffiffiffi The combination of inclusive R data in the region the need to estimate these structures using a Breit-Wigner 1.937 ≤ s ≤ 4.50 GeV. For comparison, the fit and the con- parametrization as was done in [7–9]. These data are shown tributing data are plotted against the estimate of pQCD, repre- in Fig. 16, together with the previously used incoherent sented by the dashed line and grey band. 114025-13 KESHAVARZI, NOMURA, and TEUBNER PHYS. REV. D 97, 114025 (2018) Fit of all inclusive R data 5 KEDR (16) BaBar Rb data (09) 4.5 BESII (09) CLEO (07) 4 BES (06) BES (02) BES (99) 3.5 MD-1 (96) R(s) Crystal Ball (88) 3 LENA (82) pQCD 2.5 2 2 3 4 5 6 7 8 9 10 11 √s [GeV] pffiffiffi FIG. 18. Compilation of inclusive data in range 1.937 ≤ s ≤ 11.2 GeV. The dashed line and surrounding grey band shows the estimate from pQCD for comparison. The yellow band represents the total uncertainty of the inclusive data combination. pffiffiffi Δα ð 2 Þ½ 1 937 ≤ ≤ 11 2 according to the percentage errors of the inclusive BES data had MZ inc; . s . GeV in this region [64]. With the new KEDR data [62,63], the ¼ 82.82 1.05: ð3:27Þ inclusive data combination is much improved, as shown in Fig. 17. In this range, the data combination results in pffiffiffi 1. Transition region between exclusive and inclusive data had;LO VP aμ ½inc:; 2.60 ≤ s ≤ 3.73 GeV ¼11.19 0.17; The transition region between the sum of exclusive states ð3:24Þ and the inclusive R-ratio data is of interest and deserves reexamination. For the sum of exclusive channels, while whereas from pQCD (with an inflated uncertainty [9]), the many measurements extend to 2 GeV or beyond, with estimate would be increasing energy the inclusion of more and more multi- hadronic final states is required to achieve a reliable pffiffiffi estimate of the total hadronic cross section. Previously, had;LO VP aμ ½pQCD; 2.60 ≤ s ≤ 3.73 GeV in [9], the sum of exclusive data was used up to 2 GeV, ¼ 10.82 0.35: ð3:25Þ which defined the transition point between the exclusive sum and the inclusive data combination. In this analysis, pffiffiffi the new KEDR data [62] contribute two data points below 1 937 ≤ ≤ 11 2 For the larger energy range . s . GeV, the 2 GeV, extending the lower boundary of the inclusive data resulting data combination is displayed in Fig. 18. As well down to 1.841 GeV (compared to 2 GeV in [9]) and as the differences observed between the data and pQCD providing an opportunity to reconsider the previous choice below the charm threshold, the data above it (unchanged concerning the data input in this region. since [9]) also show a slight variation from the prediction of From the lower boundary of the KEDR measurement up had;LO VP pQCD. Considering that with the new, precise KEDR data, to 2 GeV, the resulting contributions to aμ from the the differences between the inclusive data and pQCD are sum of exclusive states, the inclusive data combination, and not as large as previously and that this work is aiming at a pQCD are given in Table I. The integrated values of the predominantly data-driven analysis, the contributions in the pffiffiffi entire inclusive data region are now estimated using the had;LO VP TABLE I. Comparison of results for aμ ½1.841 ≤ s ≤ inclusive data alone. Hence, for this analysis, the contri- 2 00 bution from the inclusive data is found to be . GeV from the different available inputs in this region. pffiffiffi had;LO VP½1 841 ≤ pffiffiffi aμ . s had;LO VP ≤ 2 00 1010 aμ ½inc; 1.937 ≤ s ≤ 11.2 GeV Input . GeV × 6 06 0 17 ¼ 43.67 0.17 0.48 0.01 0.44 Exclusive sum . . Inclusive data 6.67 0.26 ¼ 43.67 0.67 ð3:26Þ pQCD 6.38 0.11 Exclusive ð<1.937 GeVÞ 6.23 0.13 þinclusive ð>1.937 GeVÞ and 114025-14 − 2 αð 2 Þ 97, MUON g AND MZ : A NEW DATA-BASED ANALYSIS PHYS. REV. D 114025 (2018) 2.8 Inclusive Low Data transition point is now 1.937 GeV, where it can be seen 2.7 Exclusive Data Combination from Fig. 19 that all available data choices at this energy are 2.6 Inclusive High Data in agreement within errors. This is further substantiated by Transition point at 1.937 GeV pQCD had;LO VP μ 2.5 KEDR (2016) Table I, where the value for a from the contribution 2.4 from exclusive states below 1.937 GeV summed with the contribution from the inclusive data combination above 2.3 R(s) 1.937 GeV is, within errors, in agreement with the 2.2 integrated values of all other choices for the data input. 2.1 Consequently, in this work, this is chosen to be the 2 transition point between the sum of exclusive states and 1.9 the inclusive R-ratio data. 1.8 ð5Þ 1.8 1.85 1.9 1.95 2 2.05 2.1 2.15 2.2 G. Total contribution of had;LO VP and Δα ð 2 Þ aμ had MZ √s [GeV] Table II lists all contributions from individual channels pffiffiffi had;LO VP FIG. 19. The energy region between 1.8 ≤ s ≤ 2.2 GeV contributing to aμ , with the corresponding total. where the fit of inclusive R ratio measurements (light blue band) From the sum of these contributions, the estimate for had;LO VP replaces the sum of exclusive hadronic final states (red band) aμ from this analysis is from 1.937 GeV to 2 GeV. The patterned light blue band and patterned red band show the continuation of the inclusive data had;LOVP ¼ð693 26 1 19 2 01 0 22 0 71 Þ aμ . . stat . sys . vp . fsr combination below 1.937 GeV and the continuation of the −10 exclusive sum above 1.937 GeV, respectively. The recent × 10 KEDR data are individually marked and included in the inclusive ¼ð693.26 2.46 Þ ×10−10; ð3:28Þ data fit. The light green band shows the data combination of old tot inclusivepffiffiffi hadronic cross section data that exist between 1 43 ≤ ≤ 2 00 where the uncertainties include all available correlations . s . GeV, which were previously discussed in 2 [9] and are not used due to their lack of precision. The estimate and local χ inflation as discussed in Sec. II B 2. Using the from pQCD is included for comparison as a dashed line with an same data compilation as described for the calculation of had;LO VP had;VP error band which is dominatedp byffiffiffi the variationpffiffiffi of the renorm- aμ , the NLO contribution to aμ is determined μ 1 μ 2 alization scale in the range 2 s < < s. here to be had;NLO VP ¼ð−9 82 0 04Þ 10−10 ð Þ inclusive data and pQCD agree within errors. However, the aμ . . × : 3:29 contribution from the sum of exclusive states disagrees with Δαð5Þ ð 2 Þ the estimates from both the inclusive data and pQCD, The corresponding result for had MZ is where the sum of exclusive states provides a smaller contribution. This is particularly visible in Fig. 19, where Δαð5Þ ð 2 Þ¼ð276 11 0 26 0 68 had MZ . . stat . sys although the sum of exclusive states agrees with the two 0 14 0 83 Þ 10−4 inclusive data points below 2 GeV at their respective . vp . fsr × ¼ð276 11 1 11 Þ 10−4 ð Þ energies, the combined sum of exclusive states is lower . . tot × ; 3:30 in general. This is largely attributed to the new data for the πþπ−π0π0 final state, where Fig. 11 shows that these new where the superscript (5) indicates the contributions from data result in a clear reduction of the fitted cross section all quark flavors except the top quark, which is added below 2 GeV.6 Due to this effect, the previous transition separately. In each case, the errors from the individual point in [9] between the sum of exclusive states and the channels and sources of uncertainty are added in quadrature inclusive data combination at 2 GeV is no longer the to determine the total error. The fractional contributions to had;LO VP preferred choice in this work, where it is clear from Fig. 19 the total mean value and uncertainty of both aμ and that these two different choices for the data input are largely Δαð5Þ ð 2 Þ had MZ from various energy intervals is shown in incompatible at this point. A more natural choice for this Fig. 20. Here, the dominance of the energy region below had;LO VP 0.9 GeV for aμ and its uncertainty is clearly evident, 6 with this being predominantly due to the contributions from Interestingly, as can be seen from Fig. 19, the sum of þ − exclusive states is in good agreement with the imprecise and, the π π channel. Notably, the pie chart depicting the ð5Þ therefore, unusedp inclusiveffiffiffi hadronic cross section data that exist fractional contributions to the ðerrorÞ2 of Δα ðM2 Þ between 1.43 ≤ s ≤ 2.00 GeV. This is in contrast with the had Z findings in the previous analyses [7–9], which observed that the reveals how the uncertainty on this quantity is dominated inclusive data in this range were lower than the sum of exclusive by the contributions from the radiative correction uncer- states. tainties. Mostly, this large error contribution comes from 114025-15 KESHAVARZI, NOMURA, and TEUBNER PHYS. REV. D 97, 114025 (2018) had;LO VP Δαð5Þ ð 2 Þ TABLE II. Summary of the contributions to aμ and had MZ calculated in this analysis. The first column indicates the hadronic final state or individual contribution, the second column gives the respective energy range of the contribution, the third column had;LO VP Δαð5Þ ð 2 Þ states the determined value of aμ , the fourth column states the determined value of had MZ , and the last column indicates any new data that have been included since [9]. The last row describes the total contribution obtained from the sum of the individual final states, with the uncertainties added in quadrature. had;LO VP 10 ð5Þ Channel Energy range [GeV] aμ × 10 Δα ð 2 Þ 104 New data had MZ × Chiralpffiffiffi perturbation theory (ChPT) threshold contributions 0 ≤ ≤ 0 600 0 12 0 01 0 00 0 00 π γ mπ psffiffiffi . . . . . πþπ− 0 87 0 02 0 01 0 00 2mπ ≤ pffiffiffis ≤ 0.305 . . . . þ − 0 3 ≤ ≤ 0 660 0 01 0 00 0 00 0 00 π π π mπ pffiffiffis . . . . . ηγ mη ≤ s ≤ 0.660 0.00 0.00 0.00 0.00 pffiffiffi pffiffiffi Data based channels ( s ≤ 1.937 GeV) 0 0 600 ≤ ≤ 1 350 4 46 0 10 0 36 0 01 π γ . psffiffiffi . . . . . [65] πþπ− 502 97 1 97 34 26 0 12 0.305 ≤ pffiffiffis ≤ 1.937 . . . . [34,35] þ − 0 0 660 ≤ ≤ 1 937 47 79 0 89 4 77 0 08 π π π . pffiffiffis . . . . . [36] πþπ−πþπ− 14 87 0 20 4 02 0 05 0.613 ≤ pffiffiffis ≤ 1.937 . . . . [40,42] þ − 0 0 0 850 ≤ ≤ 1 937 19 39 0 78 5 00 0 20 π π π π . pffiffiffis . . . . . [44] ð2πþ2π−π0Þ 1 013 ≤ ≤ 1 937 0.99 0.09 0.33 0.03 noη . pffiffiffis . 3πþ3π− 0 23 0 01 0 09 0 01 1.313 ≤ pffiffiffis ≤ 1.937 . . . . [66] ð2πþ2π−2π0Þ 1 322 ≤ ≤ 1 937 1.35 0.17 0.51 0.06 noηω . pffiffiffis . þ − 23 03 0 22 3 37 0 03 K K 0.988 ≤ pffiffiffis ≤ 1.937 . . . . [45,46,49] 0 0 1 004 ≤ ≤ 1 937 13 04 0 19 1 77 0 03 KSKL . pffiffiffis . . . . . [50,51] π 2 71 0 12 0 89 0 04 KK 1.260 ≤ pffiffiffis ≤ 1.937 . . . . [53,54] 2π 1 93 0 08 0 75 0 03 KK 1.350 ≤ pffiffiffis ≤ 1.937 . . . . [50,53,55] ηγ 0 70 0 02 0 09 0 00 0.660 ≤ psffiffiffi ≤ 1.760 . . . . [67] ηπþπ− 1 29 0 06 0 39 0 02 1.091 ≤ pffiffiffis ≤ 1.937 . . . . [68,69] ðηπþπ−π0Þ 1 333 ≤ ≤ 1 937 0 60 0 15 0 21 0 05 noω . pffiffiffis . . . . . [70] η2πþ2π− 0 08 0 01 0 03 0 00 1.338 ≤ pffiffiffis ≤ 1.937 . . . . ηω 0 31 0 03 0 10 0 01 1.333 ≤ psffiffiffi ≤ 1.937 . . . . [70,71] 0 0 0 920 ≤ ≤ 1 937 0 88 0 02 0 19 0 00 ωð→ π γÞπ . pffiffiffis . . . . . [72,73] ηϕ 0 42 0 03 0 15 0 01 1.569 ≤ pffiffiffis ≤ 1.937 . . . . ϕ → 0 04 0 04 0 01 0 01 unaccounted 0.988 ≤ psffiffiffi ≤ 1.029 . . . . 0 1 550 ≤ ≤ 1 937 0 35 0 09 0 14 0 04 ηωπ . pffiffiffis . . . . . [74] ηð→ Þ ¯ 1 569 ≤ ≤ 1 937 0 01 0 02 0 00 0 01 npp KKnoϕ→KK¯ . pffiffiffis . . . . . [53,75] ¯ 0 03 0 00 0 01 0 00 pp 1.890 ≤ pffiffiffis ≤ 1.937 . . . . [76] nn¯ 1.912 ≤ s ≤ 1.937 0.03 0.01 0.01 0.00 [77] pffiffiffi pffiffiffiEstimated contributions ( s ≤ 1.937 GeV) ðπþπ−3π0Þ 1 013 ≤ ≤ 1 937 0.50 0.04 0.16 0.01 noη . pffiffiffis . ðπþπ−4π0Þ 1 313 ≤ ≤ 1 937 0.21 0.21 0.08 0.08 noη . pffiffiffis . 3π 0 03 0 02 0 02 0 01 KK 1.569 ≤ pffiffiffis ≤ 1.937 . . . . ωð→ Þ2π 0 10 0 02 0 03 0 01 npp 1.285 ≤ pffiffiffis ≤ 1.937 . . . . ωð→ Þ3π 0 17 0 03 0 06 0 01 npp 1.322 ≤ psffiffiffi ≤ 1.937 . . . . ωð→ Þ 0 00 0 00 0 00 0 00 npp KK 1.569 ≤ psffiffiffi ≤ 1.937 . . . . ηπþπ−2π0 1.338 ≤ s ≤ 1.937 0.08 0.04 0.03 0.02 pffiffiffi pffiffiffi Other contributions ( s > 1.937 GeV) Inclusive channel 1.937 ≤ s ≤ 11.199 43.67 0.67 82.82 1.05 [56,62,63] J=ψ 6.26 0.19 7.07 0.22 ψ 0 1.58 0.04 2.51 0.06 ϒð1S − 4SÞ pffiffiffi 0.09 0.00 1.06 0.02 pQCD 11.199 ≤ s ≤∞ 2.07 0.00 124.79 0.10 pffiffiffi Total mπ ≤ s ≤∞ 693.26 2.46 276.11 1.11 114025-16 − 2 αð 2 Þ 97, MUON g AND MZ : A NEW DATA-BASED ANALYSIS PHYS. REV. D 114025 (2018) H. Comparison with the HLMNT11 evaluation To understand further how the changes in the data had;LO VP combination/input have altered the estimate of aμ and its uncertainty, a comparison of the results from this analysis and the previous HLMNT11 evaluation [9] is particularly interesting. Table III gives a channel-by- channel comparison of the two works, highlighting the differences in the individual contributions for each channel and the total sum over their respective energy ranges.7 The largest difference occurs in the πþπ− channel, where the mean value in this work is lower by almost 1σ of the HLMNT11 analysis and the uncertainty has reduced by approximately one-third. As described in the in-depth discussion of the 2π contribution in Sec. III A, this is largely due to the new, precise and highly correlated radiative return data from KLOE and BESIII and the capability of the new data combination method to utilize χ2 the correlations to their full capacity. The global min=d:o:f: of the leading and major subleading channels in this work are compared to those from the HLMNT11 analysis [9] in χ2 Table IV. The reduction of the global min=d:o:f: for the πþπ− channel further highlights that the data combination FIG. 20. Pie charts showing the fractional contributions to the for this channel has improved. The energy dependent total mean value (left pie chart) and ðerrorÞ2 (right pie chart) of changes in the resonance region are shown in Fig. 23, had;LO VP Δαð5Þ ð 2 Þ both aμ (upper panel) and had MZ (lower panel) where it can be seen that, as expected from the comparison from various energy intervals. The energy intervals for of the πþπ− results in Table III, the KNT18 data combi- had;LO VP aμ are defined by the boundaries mπ, 0.6, 0.9, 1.43, nation is in good agreement with the HLMNT11 analysis, ð5Þ 2.0, and ∞GeV. For Δα ðM2 Þ, the intervals are defined by but sits lower overall. had Z þ − the energy boundaries mπ, 0.6, 0.9, 1.43, 2.0, 4.0, 11.2, and The K K channel shows tension with the HLMNT11 ∞GeV. In both cases, the ðerrorÞ2 includes all experimental analysis, where the new data in this channel from BABAR uncertainties (including all available correlations) and local [45] and CMD-3 [46] have incurred a large increase in the χ2 ð Þ2 min=d:o:f: inflation. The fractional contribution to the error mean value, while also improving the uncertainty despite from the radiative correction uncertainties are shown in black χ2 the small increase in global min=d:o:f: This is also the case “ ” þ − þ − and indicated by rad. for the π π π π channel. Other tensions include the 0 0 ηπþπ− ηω ωð→ π0γÞπ0 KSKL, , , and channels, where again, the new, more precise data have resulted in changes outside the uncertainty due to possible FSR applied to the combi- the quoted HLMNT11 uncertainties. The KK2π channel nation of inclusive data above 1.937 GeV discussed in exhibits a similar change as discussed in Sec. III E. Sec. II A 3. This, in particular, highlights the differences in All other channels are in good agreement between the the kernel functions of the respective dispersion integrals different analyses. It it important to note that this had;LO VP ð5Þ 2 for aμ and Δα ðM Þ, where contributions from work includes three channels that were not included as part had Z ðηπþπ−π0Þ ηωπ0 Δαð5Þ ð 2 Þ of the HLMNT11 analysis: noω, , and higher energies have a larger influence on had MZ than had;LO VP μ on a . If, instead of a data driven analysis, the region 7 had;LO VP above 1.937 GeV was estimated using pQCD, it would Note that the results for individual contributions to aμ from this work that are listed in Table III differ from those effectively eliminate the impacting radiative correction given earlier in Sec. III and in Table II, as for a comparison with had;LO VP uncertainties in this region. Figure 21 shows the contribu- HLMNT11 [9], contributions to aμ from exclusive chan- tions from all hadronic final states to the hadronic R-ratio nels are evaluated up to 2 GeV. However, to consistently compare had;LO VP and its uncertainty below 1.937 GeV. Here, the individual the final results for aμ between the two works, the total final states are displayed separately as well as with the KNT18 result given in Table III is not determined as the sum of the individual contributions listed above it, but is resulting total hadronic R-ratio. The full compilation for the had;LO VP hadronic R-ratio is shown in Fig. 22. The data vector and the final result for aμ calculated in this work using the exclusive channels evaluated up to 1.937 GeV. Summing corresponding covariancepffiffiffi matrix of the hadronic R-ratio in the KNT18 values listed in Table III (i.e. choosing to evaluate the range mπ ≤ s ≤ 11.1985 GeV determined in this the sum of exclusive states from this work up to 2 GeV), results in had;LO VP −10 work is available upon request from the authors. aμ ¼ð693.06 2.45Þ × 10 . 114025-17 KESHAVARZI, NOMURA, and TEUBNER PHYS. REV. D 97, 114025 (2018) 100 Full hadronic R ratio π+π− π+π−π0 10 K+K− π+π−π0π0 π+π−π+π− 0 0 1 K S K L π0γ KKππ KKπ 0.1 π+π−π+π−π0π0 ( )no η ηπ+π− + − + − 0 R(s) π π π π π ( )no η 0.01 ωπ0 ηγ All other states π+π−π0π0π0 0.001 ( )no η ωηπ0 ηω 0.0001 π+π−π+π−π+π− π+π−π0π0π0π0 ( )no η 1e−05 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 √s [GeV] (a) 0.1 Full hadronic R ratio π+π− 0.09 π+π−π0 K+K− 0.08 π+π−π0π0 π+π−π+π− 0 0 K S K L 0.07 π0γ KKππ 0.06 KKπ π+π−π+π−π0π0 ( )no η 0.05 ηπ+π− (π+π−π+π−π0) dR(s) no η ωπ0 0.04 ηγ All other states π+π−π0π0π0 0.03 ( )no η ωηπ0 0.02 ηω π+π−π+π−π+π− π+π−π0π0π0π0 0.01 ( )no η 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 √s [GeV] (b) FIG. 21. Contributions to the total hadronic R-ratio from the different final states (upper panel) and their uncertainties (lower panel) below 1.937 GeV. The full R-ratio and its uncertainty is shown in light blue in each plot, respectively. Each final state is included as a had;LO VP new layer on top in decreasing order of the size of its contribution to aμ . J/ψ 10000 Υ ⎩ ⎧ (1s−6s)⎨ ψ(2s) 1000 Transition point at 1.937 GeV 100 φ R(s) ρ/ω 10 1 Exclusive data Inclusive data 0.1 1 10 √s [GeV] pffiffiffi FIG. 22. The resulting hadronic R-ratio shown in the range mπ ≤ s ≤ 11.1985 GeV, where the prominent resonances are labeled. 114025-18 − 2 αð 2 Þ 97, MUON g AND MZ : A NEW DATA-BASED ANALYSIS PHYS. REV. D 114025 (2018) had;LO VP TABLE III. Comparison of the contributions to aμ calculated in the HLMNT11 analysis [9] and in this work (KNT18), where had;LO VP 10 all results are given in units of aμ × 10 . The first column indicates the final state or individual contribution, the second column gives the KNT18 estimate, the third column states the HLMNT11 estimate, and the last column gives the difference between the two evaluations. Channel This work (KNT18) HLMNT11 [9] Difference Chiral perturbation theory (ChPT) threshold contributions π0γ 0.12 0.01 0.12 0.01 0.00 πþπ− 0.87 0.02 0.87 0.02 0.00 πþπ−π0 0.01 0.00 0.01 0.00 0.00 ηγ 0.00 0.00 0.00 0.00 0.00 pffiffiffi Data based channels ( s ≤ 2 GeV) π0γ 4.46 0.10 4.54 0.14 −0.08 πþπ− 502.99 1.97 505.77 3.09 −2.78 πþπ−π0 47.82 0.89 47.51 0.99 0.31 πþπ−πþπ− 15.17 0.21 14.65 0.47 0.52 πþπ−π0π0 19.80 0.79 20.37 1.26 −0.57 ð2πþ2π−π0Þ 1 08 0 10 1 20 0 10 −0 12 noη . . . . . 3πþ3π− 0.28 0.02 0.28 0.02 0.00 ð2πþ2π−2π0Þ 1 60 0 20 1 80 0 24 −0 20 noηω . . . . . KþK− 23.05 0.22 22.15 0.46 0.90 0 0 13 05 0 19 13 33 0 16 −0 28 KSKL . . . . . KKπ 2.80 0.12 2.77 0.15 0.03 KK2π 2.42 0.09 3.31 0.58 −0.89 ηγ 0.70 0.02 0.69 0.02 0.01 ηπþπ− 1.32 0.06 0.98 0.24 0.34 ðηπþπ−π0Þ 0 63 0 15 noω . . 0.63 η2πþ2π− 0.11 0.02 0.11 0.02 0.00 ηω 0.31 0.03 0.42 0.07 −0.11 ωð→ π0γÞπ0 0.88 0.02 0.77 0.03 0.11 ηϕ 0.45 0.04 0.46 0.03 −0.01 ϕ → unaccounted 0.04 0.04 0.04 0.04 0.00 ηωπ0 0.42 0.10 0.42 ηð→ Þ ¯ 0 01 0 01 npp KKnoϕ→KK¯ . . 0.01 pp¯ 0.07 0.00 0.06 0.00 0.01 nn¯ 0.06 0.01 0.07 0.02 −0.01 pffiffiffi Estimated contributions ( s ≤ 2 GeV) ðπþπ−3π0Þ 0 53 0 05 0 60 0 05 −0 07 noη . . . . . ðπþπ−4π0Þ 0 25 0 25 0 28 0 28 −0 03 noη . . . . . KK3π 0.08 0.03 0.08 0.04 0.00 ωð→ nppÞ2π 0.10 0.02 0.11 0.02 −0.01 ωð→ nppÞ3π 0.20 0.04 0.22 0.04 −0.02 ωð→ nppÞKK 0.01 0.00 0.01 0.00 0.00 ηπþπ−2π0 0.11 0.05 0.11 0.06 0.00 pffiffiffi Other contributions ( s > 2 GeV) Inclusive channel 41.27 0.62 41.40 0.87 −0.13 J=ψ 6.26 0.19 6.24 0.16 0.02 ψ 0 1.58 0.04 1.56 0.05 0.02 ϒð1S − 4SÞ 0.09 0.00 0.10 0.00 −0.01 pQCD 2.07 0.00 2.06 0.00 0.01 Total 693.26 2.46 694.91 4.27 −1.64 ηð→ ð ÞÞ ¯ had;LO VP nonpurely pionic npp KKnoϕ→KK¯ , where these final aμ has decreased between the HLMNT11 analysis states were previously unmeasured by experiment and not and this work, although this decrease is well within the estimated through isospin relations. Overall, due to the large uncertainty. In total, the uncertainty has been reduced by reduction in the πþπ− channel, it is found that the estimate of ∼42% with respect to the HLMNT11 analysis. 114025-19 KESHAVARZI, NOMURA, and TEUBNER PHYS. REV. D 97, 114025 (2018) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi χ2 ¯ TABLE IV. Comparison of the global min=d:o:f: for the this (with inflated errors for the pQCD data below the cc leading and major subleading channels determined in the threshold). As such, the estimates from this work in HLMNT11 analysis [9] and in this work (KNT18). The first Table V have been recalculated to mimic the chosen column indicates the final state or individual contribution, the energy regions of the DHMZ analysis and allow for a second column gives the KNT18 value, and the third column consistent comparison. states the HLMNT11 value. had;LO VP When comparing the total estimate of aμ from the Channel This work (KNT18) HLMNT11 [9] two analyses, the results seem to be in very good agree- ment. However, as can be seen from Table V, this masks πþπ− 1.3 1.4 πþπ−π0 2.1 3.0 much larger differences in the estimates from individual πþπ−πþπ− 1.8 1.7 channels. The most striking of these is the estimate for the πþπ− πþπ−π0π0 2.0 1.3 channel, where there is a tension of slightly more ð2πþ2π−π0Þ than 1σ between the KNT18 and DHMZ17 results. This is noη 1.0 1.2 ð2πþ2π−2π0Þ unexpected when considering the data input for both noηω 3.5 4.0 KþK− 2.1 1.9 analyses are likely to be similar and, therefore, points to K0K0 0.8 0.8 marked differences in the way the data are combined. The S L higher value of the DHMZ17 estimate seems to suggest that their data combination favors the data from the BABAR I. Comparison with other similar works measurement, with this data set being the only single set that could influence the mean value of the πþπ− channel The DHMZ group has recently released a new had;LO VP to be as high. This behavior is similar to the result estimate of aμ [78] which, due to a similar data obtained from combining the πþπ− data using only a input, is directly comparable with this work and provides simple weighted average as discussed in Sec. III A.In insight into how choices with regards to the data turn, this effect is compensated by other major subleading combination can affect results. In particular, with the final states having larger estimates in this work compared had;LO VP uncertainties of aμ from both the KNT18 and the to the DHMZ17 analysis. Specifically, the πþπ−π0, DHMZ17 analyses now less than 0.5%, it is important πþπ−πþπ− 0 0 ,andK KL estimates are noticeably lower that these differences are understood in order to quantify S in the DHMZ17 analysis. Inp addition,ffiffiffi there is tension in the reliability of different approaches and results. In [78], the region between 1.8 ≤ s ≤ 3.7 GeV, where the the authors provide a channel-by-channel breakdown of choice to use data in this region has a higher integrated their estimates for the different finals states, which are had;LO VP contribution to aμ than the DHMZ17 estimate from compared to the respective estimates from this work in pQCD. This is particularly significant when reconsidering Table V. For the exclusive data channels, the DHMZ Fig. 19, where it was observed that the sum of exclusive group chooses to take the contributions from these data pffiffiffi states from in the range 1.8 ≤ s ≤ 2.0 GeV has a cross up to 1.8 GeV, relying on estimates from pQCD above section that is lower than the estimate from pQCD. The differences seen in Table V above 1.8 GeV are then caused by cross section data below the charm production 0.1 1400 KNT18 threshold being higher than pQCD (see Fig. 17) and lower HLMNT11 1200 than pQCD above it (see Fig. 18). It should be noted that σ0(e+e- → π+π-) ωð→ Þ3π 0.05 the estimate for the npp final state from isospin 1000 had;LO VP relations, although only a small contribution to aμ , ) [nb] ) - 1 - 800 π + exhibits a significant difference between the two analyses, π 0 → suggesting a different relation has been used in the - HLMNT11 0 600 e σ + DHMZ17 analysis than in this work. 0 / (e σ ( 400 0 As well as the DHMZ analysis, an updated work by σ -0.05 Jegerlehner (FJ17) [79] resulted in an estimate of 200 had;LO VP ¼ð688 07 4 14Þ 10−10 aμ;FJ17 . . × based on the avail- -0.1 0 þ − 0.6 0.65 0.7 0.75 0.8 0.85 0.9 able e e data. Within errors, this result is in agreement √s [GeV] with both this work and the DHMZ17 analysis. Interestingly, unlike the comparison with DHMZ17, the þ − pffiffiffi FIG. 23. The normalized difference of the clusters of the π π two-pion contribution in the energy range 0.316 ≤ s ≤ þ − data fit from the KNT18 analysis with respectpffiffiffi to those from the 2.0 GeV is found in the FJ17 analysis to be aπ π ¼ 0 6 ≤ ≤ 0 9 μ;FJ17 HLMNT11 analysis in the range . s . GeV. The ð502 16 2 44Þ 10−10 width of the yellow band represents the total uncertainties of . . × [80], which is in good agreement the clusters of the HLMNT11 analysis. The πþπ− cross section is with the KNT18 estimate in the same energy range of πþπ− ¼ð501 68 1 71Þ 10−10 displayed for reference. aμ;KNT18 . . × . However, a more 114025-20 − 2 αð 2 Þ 97, MUON g AND MZ : A NEW DATA-BASED ANALYSIS PHYS. REV. D 114025 (2018) had;LO VP TABLE V. Comparison of the contributions to aμ calculated by DHMZ17 and in this work (KNT18), where had;LO VP 10 all results are given in units aμ × 10 . The first column indicates the final state or individual contribution, the second column gives the KNT18 estimate, the third column states the DHMZ17 estimate, and the last column gives the difference between the two evaluations. For the final states in this work that have low energy contributions estimated from chiral perturbation theory (see [7]), the contributions from these regions have been added to the contributions from the respective data. Channel This work (KNT18) DHMZ17 [78] Difference pffiffiffi Data based channels ( s ≤ 1.8 GeV) π0γ (data þ ChPT) 4.58 0.10 4.29 0.10 0.29 πþπ− (data þ ChPT) 503.74 1.96 507.14 2.58 −3.40 πþπ−π0 (data þ ChPT) 47.70 0.89 46.20 1.45 1.50 πþπ−πþπ− 13.99 0.19 13.68 0.31 0.31 πþπ−π0π0 18.15 0.74 18.03 0.54 0.12 ð2πþ2π−π0Þ 0 79 0 08 0 69 0 08 noη . . . . 0.10 3πþ3π− 0.10 0.01 0.11 0.01 −0.01 ð2πþ2π−2π0Þ 0 77 0 11 0 72 0 17 noηω . . . . 0.05 KþK− 23.00 0.22 22.81 0.41 0.19 0 0 13 04 0 19 12 82 0 24 KSKL . . . . 0.22 KKπ 2.44 0.11 2.45 0.15 −0.01 KK2π 0.86 0.05 0.85 0.05 0.01 ηγ (data þ ChPT) 0.70 0.02 0.65 0.02 0.05 ηπþπ− 1.18 0.05 1.18 0.07 0.00 ðηπþπ−π0Þ 0 48 0 12 0 39 0 12 noω . . . . 0.09 η2πþ2π− 0.03 0.01 0.03 0.01 0.00 ηω 0.29 0.02 0.32 0.03 −0.03 ωð→ π0γÞπ0 0.87 0.02 0.94 0.03 −0.07 ηϕ 0.33 0.03 0.36 0.03 −0.03 ϕ → unaccounted 0.04 0.04 0.05 0.00 −0.01 ηωπ0 0.10 0.05 0.06 0.04 0.04 ηð→ Þ ¯ 0 00 0 01 0 01 0 01 −0 01 a npp KKnoϕ→KK¯ . . . . . pffiffiffi Estimated contributions ( s ≤ 1.8 GeV) ðπþπ−3π0Þ 0 40 0 04 0 35 0 04 noη . . . . 0.05 ðπþπ−4π0Þ 0 12 0 12 0 11 0 11 noη . . . . 0.01 KK3π −0.02 0.01 −0.03 0.02 0.01 ωð→ nppÞ2π 0.08 0.01 0.08 0.01 0.00 ωð→ nppÞ3π 0.10 0.02 0.36 0.01 −0.26 ωð→ nppÞKK 0.00 0.00 0.01 0.00 −0.01 ηπþπ−2π0 0.03 0.01 0.03 0.01 0.00 Other contributions J=ψ 6.26 0.19 6.28 0.07 −0.02 ψ 0 1.58 0.04 1.57 0.03 0.01 ϒð1S − 4SÞ 0.09 0.00 0.09b pffiffiffi Contributions by energy region 34 54 0 56 33 45 0 65 c 1.8 ≤ psffiffiffi ≤ 3.7 GeV . . (data) . . (pQCD) 1.09 7 33 0 11 7 29 0 03 3.7 ≤ psffiffiffi ≤ 5.0 GeV . . (data) . . (data) 0.04 6 62 0 10 6 86 0 04 −0 24 5.0 ≤ psffiffiffi ≤ 9.3 GeV . . (data) . . (pQCD) . 1 12 0 01 þ 1 21 0 01 −0 09 9.3 ≤ psffiffiffi≤ 12.0 GeV . . (data pQCD) . . (pQCD) . 12.0 ≤ s ≤ 40.0 GeV 1.64 0.00 (pQCD) 1.64 0.00 (pQCD) 0.00 >40.0 GeV 0.16 0.00 (pQCD) 0.16 0.00 (pQCD) 0.00 Total 693.3 2.5 693.1 3.4 0.2 a DHMZ have not removed the decay of η to pionic states which incurs a double countingp offfiffiffi this contribution with the KKnπ channels. bDHMZ include the contributions from the ϒ resonances in the energy region 9.3 ≤ s ≤ 12.0 GeV. cDHMZ have inflated errors to account for differences between data and pQCD. 114025-21 KESHAVARZI, NOMURA, and TEUBNER PHYS. REV. D 97, 114025 (2018) DEHZ03: 696.3 ± 7.2 available parametrizations [4] instead of using the available HMNT03: 692.4 ± 6.4 data. A comparison of recent and previous evaluations of had;LO VP þ − DEHZ06: 690.9 ± 4.4 aμ determined from e e → hadrons cross section 8 HMNT06: 689.4 ± 4.6 data is shown in Fig. 24, which highlights the agreement FJ06: 692.1 ± 5.6 between the different works and the improvement in the DHMZ10: 692.3 ± 4.2 precision of the respective analyses. JS11: 690.8 ± 4.7 HLMNT11: 694.9 ± 4.3 J. SM prediction of g − 2 of the muon FJ17: 688.1 ± 4.1 DHMZ17: 693.1 ± 3.4 The SM prediction of the anomalous magnetic moment KNT18: 693.3 ± 2.5 of the muon is determined by summing the contributions from all sectors of the SM, such that 685 690 695 700 705 710 715 had, LO VP 10 aμ x 10 SM QED EW had aμ ¼ aμ þ aμ þ aμ ; ð3:31Þ FIG. 24. Comparison of recent and previous evaluations of had;LO VP þ − aμ determined from e e → hadrons cross section data. QED EW where aμ is the QED contribution, aμ is the electro- The analyses listed in chronological order are DEHZ03 [81], had weak contribution, and aμ are the hadronic contributions HMNT03 [7], DEHZ06 [82], HMNT06 [8], FJ06 [83], DHMZ10 [84], JS11 [85], HLMNT11 [9], FJ17 [79], and DHMZ17 [78]. due to the hadronic vacuum polarization and hadronic light- The prediction from this work is listed as KNT18, which defines by-light scattering [see Eq. (1.2)]. The QED contributions the uncertainty band that the other analyses are compared to. are known up to and including five-loop accuracy. The five- loop calculation has recently been completed numerically detailed comparison with the estimates of other channels by Kinoshita et al. [86,87] to evaluate all 12,762 five-loop determined in [79,80] is not possible as the FJ17 analysis diagrams. This calculation includes all contributions that chooses to estimate certain resonance contributions using are due to photons and leptons alone. They are found to be QED −10 aμ ¼ 11 658 471.8971 ð0.0007Þð0.0017Þð0.0006Þð0.0072Þ × 10 ¼ð11 658 471.8971 0.007Þ × 10−10; ð3:32Þ where the uncertainties are owing to the uncertainty on the used in [9]. Although a relatively small contribution when QED lepton masses, the four-loop contributions, the five-loop compared to aμ , the uncertainty is not negligible con- contributions, and the determination of α using measure- sidering the projected experimental accuracy, but is small ments of 87Rb, respectively. With such a precise determi- when compared to the hadronic uncertainties. However, QED nation of aμ resulting from a perturbative series that with this contribution known safely to two-loop accuracy, converges extremely well, the QED result seems stable. It the electroweak estimate is also very well under control. should be noted, however, that the four-loop and five-loop For the hadronic vacuum polarization contributions, the contributions rely heavily on numerical integrations and leading order and next-to-leading order contributions have independent checks of these results are crucial. This has been calculated in this work. The LO contribution, from had;LO VP recently been accomplished through several different analy- Eq. (3.28), was found to be aμ ¼ð693.26 2.46Þ × −10 had;NLO VP ses [88–94], which corroborate the results from Kinoshita 10 and the NLO was given in Eq. (3.29) as aμ ¼ and collaborators. Therefore, it is safe to assume that the ð−9.82 0.04Þ × 10−10. The calculation of the NNLO estimate for the QED contribution is well under control. hadronic vacuum polarization contribution has been The contribution from the EW sector is well known to achieved for the first time in [10] (see also the evaluation two-loop accuracy [95–99]. With the mass of the Higgs in [79]), where contributions from five individual classes of now known, the updated estimate [100] gives diagrams (each with an independent, respective kernel had;NNLO VP μ ¼ EW −10 function) were determined. It was found to be a aμ ¼ð15.36 0.10Þ × 10 ; ð3:33Þ −10 ð1.24 0.01Þ × 10 . This estimate is slightly larger than where the knowledge of the Higgs mass has halved the expected and results in a slight increase to the mean value of had;VP uncertainty of this contribution compared to the estimate aμ . Summing these, the total contribution to the anoma- lous magnetic moment from the hadronic vacuum polariza- 8 tion is In addition, an evaluation using a global fit based on the broken hidden local symmetry model [31] prefers lower values had;VP −10 had;LO VP μ ¼ð684 68 2 42Þ 10 ð Þ for aμ than those displayed in Fig. 24. a . . × : 3:34 114025-22 − 2 αð 2 Þ 97, MUON g AND MZ : A NEW DATA-BASED ANALYSIS PHYS. REV. D 114025 (2018) It should be noted that the negative NLO contribution results in an anticorrelation between its uncertainty and the uncer- DHMZ10 tainty from the LO contribution, consequently resulting in a JS11 slight reduction in the overall uncertainty that has been HLMNT11 incorporated into Eq. (3.34). FJ17 The hadronic LbL contributions, although small compared to the hadronic vacuum polarization sector, have, in the past, DHMZ17 been determined through model-dependent approaches. These KNT18 are based on meson exchanges, the large N limit, ChPT BNL c 3.7σ estimates, short distance constraints from the operator product BNL (x4 accuracy) expansion, and pQCD. Over time, several different approaches 7.0σ had;LbL to evaluating aμ have been attempted, resulting in good 160 170 180 190 200 210 220 0 π SM 10 agreement for the leading Nc ( exchange) contribution, but (aμ x 10 )−11659000 differing for subleading effects. A commonly quoted deter- SM mination of the LbL contribution is the “Glasgow consensus” FIG. 25. A comparison of recent and previous evaluations of aμ . had;LbL estimate of aμ ðGlasgow consensusÞ¼ð10.5 2.6Þ × The analyses listed in chronological order are DHMZ10 [84],JS11 10−10 [101] (alternatively, see [102–105]). However, recent [85], HLMNT11 [9],FJ17[79], and DHMZ17 [78]. The prediction – had;LbL from this work is listed as KNT18, which defines the uncertainty works [106 108] have reevaluated the contribution to aμ band that other analyses are compared to. The current uncertainty due to axial exchanges, where it has been found that this on the experimental measurement [1–4] is given by the light blue contribution has, in the past, been overestimated due to an band. The light grey band represents the hypothetical situation of incorrect assumption that the form factors for the axial meson the new experimental measurement at Fermilab yielding the same exp contribution are symmetric under the exchange of two photon mean value for aμ as the BNL measurement, but achieving the momenta [106]. Under this assumption, the determination in projected fourfold improvement in its uncertainty [5]. [102] previously found the axial vector contribution to be had;LbL;axial −10 had;LbL aμ ¼ð2.2 0.5Þ×10 . Correcting this reduces this determining aμ using dispersive approaches are also had;LbL;axial −10 contribution to aμ ¼ð0.8 0.3Þ×10 [106,107]. very promising [111–116], where the dispersion relations are Applying this adjustment to the Glasgow consensus result, the formulated to calculate either the general hadronic LbL tensor had;LbL estimate in [108] finds or to calculate aμ directly. These approaches will allow for the determination of the hadronic LbL contributions from had;LbL −10 aμ ¼ð9.8 2.6Þ × 10 ; ð3:35Þ experimental data and, at the very least, will invoke stringent had;LbL constraints on future estimates. Last, there has been huge aμ which is the chosen estimate for in this work. This progress in developing methods for a direct lattice simulation result is notably lower than the previously accepted LbL had;LbL – SM of aμ [110,117 123]. With a proof of principle already estimates and will incur an overall downward shift on aμ .It well established, an estimate of approximately 10% accuracy is, however, still within the original uncertainties when seems possible in the near future. Considering these develop- comparing with the original Glasgow consensus estimate. ments and the efforts of the Muon g − 2 Theory Initiative Alternatively, it should be noted that the estimate of had;LbL −10 [124] to promote the collaborative work of many different aμ ¼ð10.2 3.9Þ × 10 [108,109], which is a result had;LbL groups, the determination of aμ on the level of the that is independent of the Glasgow consensus estimate, Glasgow consensus will, at thevery least, be consolidated and could be employed here. In addition, the recent work [105] a reduction of the uncertainty seems highly probable on the has provided an estimate for the next-to-leading order − 2 had;NLO-LbL time scales of the new g experiments. hadronic LbL contribution. It has found aμ ¼ −10 Following Eq. (3.31), the sum of all the sectors of the SM ð0.3 0.2Þ × 10 , which does not alter the hadronic results in a total value of the anomalous magnetic moment LbL contribution significantly, but is taken into account of the muon of in the full SM prediction given below. SM ¼ð11659182 04 3 56Þ 10−10 ð Þ Much work has also been directed at the possibility of a aμ . . × ; 3:36 had;LbL model independent calculation of aμ to further consoli- where the uncertainty is determined from the uncertainties date the SM prediction of aμ. One approach involves the of the individual SM contributions added in quadrature. measurement of transition form factors by KLOE-2 and Comparing this with the current experimental measurement BESIII, which can be expected to constrain the leading given in Eq. (1.1) results in a deviation of 0 0 pseudoscalar-pole (π , η; η ) contribution to a precision of −10 Δaμ ¼ð27.06 7.26Þ × 10 ; ð3:37Þ approximately 15% [108]. Alternatively, the pion transition form factor (π0 → γ γ ) can be calculated on the lattice for the corresponding to a 3.7σ discrepancy. This result is compared SM same purpose [110]. New efforts into the prospects of with other determinations of aμ in Fig. 25. In particular, a 114025-23 KESHAVARZI, NOMURA, and TEUBNER PHYS. REV. D 97, 114025 (2018) had;VP TABLE VI. Comparison of recent and previous evaluations of anomalous magnetic moment of the muon, aμ , and the Δαð5Þ ð 2 Þ þ − → hadronic contributions to the effective QED coupling at the had MZ determined from e e hadrons cross section data α−1ð 2 Þ Δα ð 2 Þ and the corresponding results for MZ . Z boson mass, had MZ . These quantities have been determined using the available eþe− → hadrons cross Analysis Δαð5Þ ð 2 Þ 104 α−1ðM2 Þ had MZ × Z section data as input into corresponding dispersion rela- DHMZ10 [84] 275.59 1.04 128.952 0.014 tions, with an aim to achieve both accurate and reliable HLMNT11 [9] 276.26 1.38 128.944 0.019 results from a predominantly data driven analysis. Since FJ17 [129] 277.38 1.19 128.919 0.022 [9], all aspects concerning the radiative corrections of the DHMZ17 [78] 276.00 0.94 128.947 0.012 data and the data combination have been reassessed in this KNT18 [This work] 276.11 1.11 128.946 0.015 work. Specifically, the data are now combined using an iterative, linear χ2-minimization developed from a method that has been advocated to be free of bias and that has been comparison with the HLMNT11 estimate given in Eq. (1.7) studied in detail. Importantly, this data combination method SM shows an improvement in the total uncertainty of aμ of allows for the full use of any available correlated statistical ∼27%. It should be noted that although, as stated above, the and systematic uncertainties, incorporating experimental had;LO VP DHMZ17 estimate for aμ [78] is lower than the value covariance matrices in the combination in a bias-free SM obtained in this work, the estimate of aμ from DHMZ17 is approach. These changes, plus the large quantity of new higher than the estimate from this analysis as DHMZ17 hadronic cross section data, have resulted in improved chooses to use the estimate for the hadronic light-by-light estimates for nearly all hadronic channels. This is particu- þ − had;LbL −10 π π contribution of aμ ¼ð10.5 2.6Þ × 10 [101]. larly true for the channel, where the precision of this final state has improved by approximately one third πþπ− K. Determination of αðM2 Þ compared to [9], with aμ from both analyses in very Z good agreement. Importantly, the inclusion of recently From Eq. (3.30), the five-flavor hadronic contribution to released neutral data in the KKπ and KK2π channels Δαð 2 Þ Δαð5Þ ð 2 Þ¼ð276 11 1 11Þ MZ is found to be had MZ . . × has removed the need to rely on isospin relations to 10−4. Combining this with the leptonic contribution estimate these final states. In addition, new inclusive Δα ð 2 Þ¼ð314 979 0 002Þ 10−4 lep MZ . . × [125,126] and the hadronic R-ratio data from the KEDR Collaboration have Δα ð 2 Þ¼ð−0 7180 improved the inclusive data combination. In particular, they contribution due to the top quark top MZ . 0.0054Þ × 10−4 [127,128], the total value of the QED have provided the opportunity to reconsider the transition coupling at the Z boson mass is found in this work to be region between the sum of exclusive states and the inclusive data, which has resulted in the transition −1 2 2 ð5Þ 2 2 −1 point being chosen to be at 1.937 GeV in this work, where α ðM Þ¼ð1 − Δα ðM Þ − Δα ðM Þ − Δα ðM ÞÞα Z lep Z had Z top Z the different choices for the data input in this point ¼ 128.946 0.015: ð3:38Þ are in agreement within errors. The complete data compi- lation has yielded the full hadronic R-ratiopffiffiffi and its covari- Here, the top contribution is determined using the top quark ance matrix in the energy range mπ ≤ s ≤ 11.2 GeV. ¼ 173 1 0 6 mass, mt . . GeV, and the value for strong Using these combined data, for the muon g − 2, this α ð Þ¼ had;LO VP −10 coupling constant at the mass of the Z boson, s MZ analysis found aμ ¼ð693.26 2.46Þ × 10 and 0.1182ð12Þ [4]. A comparison of these results with other had;NLO VP −10 aμ ¼ð−9.82 0.04Þ × 10 . This has resulted in Δαð5Þ ð 2 Þ α−1ð 2 Þ determinations of had MZ and MZ is given in a new estimate for the Standard Model prediction of SM −10 Table VI. Note that, as discussed in Sec. III G, the different aμ ¼ð11659182.04 3.56Þ × 10 , which deviates Δαð5Þ ð 2 Þ 3 7σ weighting of the kernel function of the had MZ dispersion from the current experimental measurement by . .For had;LO VP integral compared to the integral for aμ results in the effective QED coupling, the new estimate in this work contributions from higher energy regions having a larger Δαð5Þ ð 2 Þ¼ð276 11 1 11Þ 10−4 of had MZ . . × yields an updated Δαð5Þ ð 2 Þ had;LO VP influence on had MZ than on aμ . Therefore, the value for the QED coupling at the Z boson mass α−1ð 2 Þ¼128 946 0 015 choice to use either the available inclusive data or pQCD of MZ . . . SM above ∼2 GeV as discussed in Sec. III I can have marked In general, the predictions for aμ have been further ð5Þ differences on the values and errors obtained for Δα ðM2 Þ. scrutinized and are well established. In particular, the had Z improvements in the uncertainty on the level discussed IV. CONCLUSIONS AND FUTURE in [130] are on track in preparation for the new exper- imental results from Fermilab and J-PARC. The efforts of PROSPECTS FOR THE MUON g − 2 the Muon g − 2 Theory Initiative and the groups involved This analysis, KNT18, has completed a full reevaluation within it show great progress and promise in improving the SM of the hadronic vacuum polarization contributions to the estimate of aμ further. Importantly, it should be noted that 114025-24 − 2 αð 2 Þ 97, MUON g AND MZ : A NEW DATA-BASED ANALYSIS PHYS. REV. D 114025 (2018) there is no indication thus far that the SM prediction does the hadronic vacuum polarization contribution that is more had;LbL not deviate with the current experimental measurement by precise than the currently quoted uncertainties for aμ . more than 3σ. For the hadronic vacuum polarization However, given these developments in improving the contributions, there is scope to further improve the esti- Standard Model prediction of aμ and the formidable þ − mates, as new measurements of the π π cross section are progress made by the new muon g − 2 experiments at planned to be released in the near future from BABAR, Fermilab and J-PARC, the prospects either to establish the CMD-3, SND, and, possibly, BELLE-2. Also, efforts are existence of new physics contributing to aμ or to rule out currently being made to measure new inclusive R-ratio data many new physics scenarios are highly compelling. by BESIII and the experiments at Novosibirsk (SND, CMD-3, KEDR). This, in particular, will benefit the ACKNOWLEDGMENTS Δαð5Þ ð 2 Þ evaluation of had MZ which, as discussed in Sec. III had;LO VP We recognize and wholeheartedly thank Alan Martin and K, is more sensitive than aμ to the precision of the Kaoru Hagiwara for their many-years-long collaboration in data from higher energy regions. In addition, lattice – had;VP [7 9], and also thank Ruofan Liao for his collaboration in determinations of aμ are rapidly improving. Recent [9]. We thank Simon Eidelman, Fedor Ignatov, Andreas work that combines data from lattice QCD with those from Nyffeler, and the DHMZ17 group (Michel Davier, Andreas experimental R-ratio data have already provided extremely Hoecker, Bogdan Malaescu, and Zhiqing Zhang) for numer- accurate results that are in good agreement with the current ous useful discussions. We also acknowledge the discus- estimates from the dispersive method [131]. Furthermore, Working Group on Radiative Corrections had;LO VP sions within the efforts to experimentally measure aμ in the spacelike and MC Generators for Low Energies (Radio region are progressing [132,133] and would provide an MonteCarLow) and The Muon g-2 Theory Initiative con- alternative check of the results from the dispersive cerning this work. The work of A. K. is supported by STFC approach. Should any or all of these advances reduce under the consolidated Grant No. ST/N504130/1. The work had;VP the uncertainty of aμ even further, the improvement of of D. N. is supported by JSPS KAKENHI Grants the hadronic light-by-light estimates will become particu- No. JP16K05323 and No. JP17H01133. The work of T. larly crucial. This is further driven by the fact that the result T. is supported by STFC under the consolidated Grants had;VP for aμ determined in this analysis is the first estimate of No. ST/L000431/1 and No. ST/P000290/1. [1] G. W. Bennett et al. 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