The $g-2$ of charged leptons, $\alpha(M_Z^2)$ and the hyperfine splitting of muonium

Following updates in the compilation of $e^+e^-\rightarrow{\rm hadrons}$ data, this work presents re-evaluations of the hadronic vacuum polarisation contributions to the anomalous magnetic moment of the electron ($a_e$), muon ($a_\mu$) and tau lepton ($a_\tau$), to the ground-state hyperfine splitting of muonium and also updates the hadronic contributions to the running of the QED coupling at the mass scale of the $Z$ boson, $\alpha(M_Z^2)$. Combining the results for the hadronic vacuum polarisation contributions with recent updates for the hadronic light-by-light corrections, the electromagnetic and the weak contributions, the deviation between the measured value of $a_\mu$ and its Standard Model prediction amounts to $\Delta a_{\mu} = (28.02 \pm 7.37) \times 10^{-10}$, corresponding to a muon $g-2$ discrepancy of $3.8\sigma$.


Introduction
For the charged leptons (l = e, µ, τ ), the study of their anomalous magnetic moment, a l = (g − 2) l /2, continues to serve as a long-standing test of the Standard Model (SM) and as a powerful indirect search of new physics. In each case, the SM prediction of the anomalous magnetic moment is determined by summing the contributions from all sectors of the SM, such that a SM The recent complete re-evaluation of the hadronic VP contributions to a µ preceding this work (denoted as KNT18) found the SM prediction to be a SM µ (KNT18) = (11 659 182.04 ± 3.56)×10 −10 [1], with the uncertainty still entirely dominated by the non-perturbative, hadronic sector. Compared with the current experimental world average of a exp µ = (11 659 209.1 ± 6.3) × 10 −10 [2][3][4][5], a discrepancy of ∆a µ = a exp µ − a SM µ = (27.06 ± 7.26) × 10 −10 was found, with the SM prediction being 3.7σ below the experimental measurement. With new efforts at Fermilab (FNAL) [6,7] (and later at J-PARC [8]) aiming to reduce the experimental uncertainty by a factor of four, coupled with the ongoing efforts of the Muon g −2 Theory Initiative [9] to improve the determination of the various SM contributions in conjunction with these new measurements, it is imperative that the determination in [1] is continuously updated and improved.
A relatively new and interesting deviation has now also arisen in the study of the electron g − 2. Until recently, the comparison of the exceptionally precise measurement of a exp e = (1 159 652 180.73 ± 0.28) × 10 −12 [10] with the SM prediction a SM e (α Rb ) = (1 159 652 182.032 ± 0.720)×10 −12 [11] (which updated [12]) deviated only at the level of 1.7σ. Here, α Rb denotes that the SM prediction has been determined using the measurement of the fine-structure constant via rubidium (Rb) atomic interferometry [13], which contributes the dominant uncertainty to this prediction of a SM e . However, the use of a new, more precise measurement of α using caesium (Cs) atomic interferometry [14] results in an estimate of a SM e (α Cs ) = (1 159 652 181.61±0.23)×10 −12 . This implies a deviation of ∆a e = a exp e − a SM e (α Cs ) = (−0.88 ± 0.36) × 10 −12 , corresponding to a 2.5σ difference. 1 This result has invoked much theoretical work into the possibility of simultaneously explaining the differences in both the electron and muon sector, which must also explain the current sign difference seen between ∆a e and ∆a µ (see e.g. [16]). Although, due to the small mass of the electron, a SM e is less sensitive to strong effects than a SM µ , the recently observed changes in the electron sector make it important that the hadronic contributions to the electron g − 2 are also updated from the previous determination in [17] (denoted here as NT12).
Measurements of the anomalous magnetic moment of the tau lepton, a exp τ , are notoriously difficult due to the short lifetime of the τ and, as such, no direct measurement of a τ has yet been achieved. Limits on a exp τ were set by the DELPHI collaboration to be −0.052 < a exp τ < 0.013 at the 95% confidence level [2,18], which is quoted in the form a exp τ = −0.018 (17) in [18]. By standard lepton mass-scaling arguments, a τ is more sensitive to heavy new physics than a µ by a factor of m 2 τ /m 2 µ ∼ 280. However, the relative contributions of strong effects compared to both the electron and the muon make a τ more sensitive to hadronic contributions by the same argument. The hadronic VP contributions were determined in [19] to be a had, VP is an interesting undertaking and may prove useful, should experimental techniques improve to be able to better probe the anomaly of the τ lepton.
It follows that this work, denoted KNT19, will update the hadronic vacuum polarisation contributions to a l = (g − 2) l /2 for all l = e, µ, τ . These are calculated utilising dispersion integrals and the experimentally measured cross section, where the superscript 0 denotes the bare cross section (undressed of all vacuum polarisation effects) and the subscript γ indicates the inclusion of effects from final state radiation (FSR) of (one or more) photons (see [1] for details). The determination of the hadronic R-ratio, defined as and obtained from the updated compilation of all available e + e − → hadrons data, is the foundation of this endeavour. Here, α = α(0) is the fine-structure constant. From this, the leading-order (LO) hadronic VP contributions to a l can be determined via the dispersion relation where s th = m 2 π and K l (s) is a well-known kernel function [20,21]. Expressed in the form K l (s) ≡ 3s/m 2 l K(s),K l (s) is a monotonically-increasing function that behaves asK l (s) → 1 as s → ∞. This behaviour differs slightly for each lepton. In the case of the electron, the deviation 1 Note that very recently there has been an independent calculation of the purely photonic five-loop contributions to ae [15], which gives a different value compared to the one in [11] and which, if adopted, would slightly change the predictions for a SM e and ∆ae. ofK e (s) from 1 is almost negligible for all s and causes a had, LO VP e to be heavily dominated by the contributions from the lowest energies [17]. For the muon, K µ (s) behaves as K µ (s) ∼ m 2 µ /(3s) at low energies and also accentuates the low energy domain [1,22], although not as heavily as for the electron. ForK τ (s), the larger τ mass results in a functional structure that further increases the role of contributions from higher energies relative toK µ (s), although the role of lower energies is still prominent [19]. At next-to-leading order (NLO), similar dispersion integrals and kernel functions exist [22,23], allowing for a had, NLO VP l to be determined in conjunction with the LO contributions. At NNLO, a had, NNLO VP l has been determined for l = e, µ [24]. In addition, the determination of the hadronic R-ratio is a crucial input for two other precision observables which test the SM. First, the hadronic contributions to the effective QED coupling ∆α (5) had (q 2 ) allow for an update of this quantity at the scale of the Z boson mass, α(M 2 Z ), which hinders the accuracy of EW precision fits. Second, the hadronic VP corrections are a nonnegligible part of the ground-state hyperfine splitting (HFS) of muonium, ∆ν Mu , which can be used to determine the electron-to-muon mass ratio and, hence, the muon mass.
This paper continues, in Section 2, with a description of the updates in the compilation of hadronic cross data since [1]. Section 3 details the new results for the contributions to a had, LO VP l for each l = e, µ, τ (with corresponding new estimates for a SM l ), followed by updated predictions for α(M 2 Z ) and ∆ν had, VP

Mu
. Conclusions and discussions of future prospects are given in Section 4.

Updates since the last analysis (KNT18)
The data combination methodology in this work is unchanged from [1] and, unless differences are explicitly stated, the cross section determination for each hadronic channel is unaltered. However, various updates with respect to the available data have been accounted for and are described in the following. As before, results for a had, LO VP µ are quoted with their respective statistical (stat) uncertainty, systematic (sys) uncertainty, vacuum polarisation (vp) correction uncertainty and final state radiation (fsr) correction uncertainty. The total (tot) uncertainty is determined from the individual sources added in quadrature.

π + π − channel
The all-important π + π − channel is modified only by the introduction of a new radiative return measurement based on data taken at the CLEO-c experiment between 0.3 ≤ √ s ≤ 1.0 GeV, covering the dominant ρ resonance region [25]. The measurement consists of two data sets: the first taken at e + e − energies at the centre-of-mass of the ψ(3770) resonance and the second at the ψ(4170) resonance. Although these measurements come already undressed of VP effects as required by equation (1.2), the undressing procedure applied in [25] used an outdated routine [26]. Therefore, in this work, the published cross section values are redressed utilising the routine provided in [26] and then undressed via the KNT18 vacuum polarisation routine, vp knt v3 0 [1,27]. 2 Notably, the statistical and systematic uncertainties of the CLEO-c data are large compared to the KLOE [28][29][30][31] and BaBar [32] measurements and, therefore, cannot resolve the tension between the KLOE and BaBar data. In addition, in the KNT19 data combination, the systematic uncertainties of the two CLEO-c data sets are taken to be 100% correlated, which further limits their influence.  The combined cross section and the dominant contributing measurements are displayed in the ρ region and magnified in the ρ-ω interference region in Figure 1. Figure 2 shows the updated comparison of the evaluations of a π + π − µ from the radiative return measurements and the combination of remaining direct scan data in the vicinity of the ρ resonance. Although the new CLEO-c data are compatible with both the KLOE and BaBar measurements, resulting in a marginal improvement in the quality of the overall fit, as expected the combination is largely unchanged due to the large uncertainties of the CLEO-c data. The tension between BaBar and KLOE persists, emanated in the KNT19 combination of all π + π − data, which is still dominated by the three KLOE cross section measurements and their precise, highly-correlated uncertainties. This is further exemplified by Figure 3, which clearly indicates the tension between KLOE and BaBar, and between the fit of all π + π − data and BaBar, especially in the high-energy tail of the ρ resonance. contributing to a π + π − µ , and the fit of all data. For comparison, the individual sets have been normalised against the fit and have been plotted in the ρ region. The green band represents the BaBar data and their errors (statistical and systematic, added in quadrature). The yellow band represents the full data combination which incorporates all correlated statistical and systematic uncertainties. However, the width of the yellow band simply displays the square root of the diagonal elements of the total output covariance matrix of the fit.
This value is entirely consistent with [1]. The mean value has increased by ∼ 25% of the previous error, which itself has reduced by only ∼ 3%. As before, tensions in the data are accounted for in the local χ 2 error inflation, increasing the uncertainty of a π + π − µ by ∼ 14%. This has decreased from ∼ 15% in [1], also reflected in the slight decrease in the global χ Although the results of this work are obtained from directly integrating the combined data, detailed analyses employing constraints based on analyticity and unitarity have been performed in [33][34][35][36][37]. These additional constraints have the potential to improve the determination of the two-pion cross section and to possibly reduce the error, especially at low energies where limited data are available. The results obtained in these works are, overall, largely compatible with the determination of this analysis, but lead to slightly larger results for a had, LO VP µ in the energy range √ s < 0.6 GeV. A detailed comparison with these values is beyond the scope of this work, but will be presented as part of the studies of the Muon g − 2 Theory Initiative [9].
2.2 π + π − π 0 channel A recent study of the three-pion contribution to the hadronic vacuum polarisation based on a global fit function using analyticity and unitarity constraints [38] highlighted major differences arising in various determinations of a π + π − π 0 µ . These were attributed to the choice of cross section  Figure 4: The cross section σ 0 (e + e − → π + π − π 0 ) in the region of the narrow ω resonance. In Figure 4 interpolation used in the prominent ω resonance region when integrating the data. Due to a lack of data and a (relatively) wide-binning in the narrow ω resonance itself, the trapezoidal rule integration used in [1,22,39,40], while consistent with the direct data integration procedure utilised in these works, led to a value of a π + π − π 0 µ in [1] larger than found in [37,38]. In order to address this issue in this work, the clusters and covariance matrix elements corresponding to the fitted ω resonance alone have been interpolated to a 0.2 MeV binning using a quintic polynomial. The newly finer-binned resonance, along with the entire π + π − π 0 cross section, are then integrated using the trapezoidal rule integral to ensure consistency with the general KNT data combination procedure applied to all other channels. This results in an improved estimate of compared to a π + π − π 0 µ (KNT18) = (47.79 ± 0.89) × 10 −10 in [1]. Figure 4(a) shows an enlargement of the ω resonance region, where the comparison between the previously used trapezoidal rule integral (black dashed line), a cubic polynomial interpolation (dashed-dotted green line) and the quintic polynomial (solid pink line) interpolation are visible, highlighting the improvement that this change has made. 3 It can also be seen here that whilst the linear interpolation clearly overestimates the resonance in the tails, the cubic interpolation seemingly underestimates and overestimates the cross section in various places in the tail, hence the choice of the quintic polynomial. The resulting KNT19 determination of the ω resonance in the π + π − π 0 channel and all contributing data are shown in Figure 4

Other channels
There have been a number of small data updates (see [41][42][43][44][45][46]) in other channels since [1]. The affected channels are all depicted in Figure 5 and Figure 6. Notably, the π 0 γ channel now includes a new measurement from the SND experiment [41], which greatly extends the previous upper border of the channel from 1.35 GeV to 1.935 GeV in this work. The changes to a π 0 γ µ are negligible, confirming that no higher energy contributions were missed previously in this hadronic mode. Two new channels are now included in the KNT19 data compilation. A measurement of the 2π + 2π − ω channel by CMD-3 [46] provides a negligibly small addition to a had, LO VP µ . This process, together with a measurement of the 2π + 2π − η mode, have provided the production mechanisms to measure the seven-pion final state 3π + 3π − π 0 in the same work [46], which is the first inclusion of a final state with more than six pions. After removing the contributions from the η and ω resonances to avoid double-counting, the 3π + 3π − π 0 channel is statistically consistent with zero below the upper energy boundary of the sum of exclusive states used here, i.e. 1.937 GeV. Once again, it is encouraging to ratify that no large contributions were missed Figure 6: The resulting cross sections of those hadronic channels contributing to the KNT19 data compilation that were previously estimated via isospin relations. In Figure 6(c), the abbreviation '→npp' represents the resonant decay to non-purely-pionic modes.
from these channels in the KNT18 data compilation. Lastly, it is important to mention that the three modes π + π − 3π 0 , π + π − 2π 0 η and ωπ 0 π 0 that were previously unmeasured have now been measured by BaBar [42]. These allow, for the first time, for their corresponding hadronic contributions to be estimated using experimental data instead of previously used isospin relations. All three channels are shown in Figure 6, where the agreement in each case between the data and the isospin prediction is good. The resulting integrated contributions to a had, LO VP µ are all consistent with the theory estimates previously given in [1].  To obtain ∆α (5) had (M 2 Z ), the data are integrated using equation (3.14) given in Section 3.4. For ∆ν had, VP Mu , equation (3.20) in Section 3.5 is used. In the following section, the KNT19 results for a e , a µ , a τ , α(M 2 Z ) and ∆ν Mu are presented separately. For each of the lepton g − 2 results, the values for the LO and NLO hadronic VP contributions as calculated in this work are given, followed by corresponding updated estimates for the respective SM predictions and any necessary discussions.
As the NNLO hadronic VP contributions are not calculated in this work, the result a had, NNLO VP e = (2.80 ± 0.01) × 10 −14 from [24] is adopted which utilises the HLMNT11 [40] data compilation for the hadronic R-ratio. 4 For the hadronic LbL contributions, the value a had, LbL where, due to the complete correlations from the same input R-ratio, the errors of the hadronic VP contributions have been added linearly. Compared to a had e (NT12) = (167.8 ± 1.4) × 10 −14 in [17], the mean value found in this work is outside the quoted error given in [17]. However, it should be noted that no determination of the NNLO hadronic VP contributions was available for [17], whereas in this work the addition of a had, NNLO VP e = (2.80 ± 0.01) × 10 −14 constitutes, similar to the case of the muon, a significant additional correction.
The EW contributions, a EW e = (3.053 ± 0.023) × 10 −14 , are also taken from [47]. For the QED contributions, there are now two options depending on the choice for the value of α. 5 As described in Section 1, the use of the measurement of α from Rb atomic interferometry [13] or Cs atomic interferometry [14] leads to an interesting comparison with a exp e . For each case, the 4 During the KNT18 analysis, the authors of [24] kindly repeated their analysis with the KNT18 data compilation and found negligible changes with respect to their published result.   [14] . The comparison of these results with the experimental measurement of a e [10] is given in Table 2 and shown in Figure 7. The values of the deviation between theory and experiment of ∆a e (α Rb ) = (−1.31 ± 0.77) × 10 −12 (1.7σ) and ∆a e (α Cs ) = (−0.89 ± 0.36) × 10 −12 (2.5σ) confirm the findings in [11] and [14], respectively.
These results are consistent with the KNT18 analysis. At LO, the integral over the hadronic Rratio determined in [1] resulted in a had, LO VP µ (KNT18) = (693.26±2.46)×10 −10 . Comparing this with equation (3.6), the reduction in the mean value comes entirely from the updated treatment of the ω resonance in the π + π − π 0 channel described in Section 2.2. This change counteracts the small increase in the mean value from the π + π − channel due to the inclusion of the CLEO-c data [25] detailed in Section 2.1, as well as the very small increase due to the newly included channels reported in Section 2.3. The marginal decrease in the overall uncertainty is also due to the inclusion of the CLEO-c data [25], which as explained previously has caused a small decrease in the local χ 2 error inflation of the dominant two-pion contribution. A comparison of this result with similar evaluations of a had, LO VP µ determined from e + e − → hadrons cross section data is shown in Figure 8. It is important to note that there is clear stability and overall agreement between the different analyses/groups over the consecutive years, despite contrasting choices the different groups have made concerning how to treat the hadronic cross section data, where to use perturbative QCD (pQCD) instead of data and the application of other possible theoretical constraints. 6 Combining the results (3.6) and (3.7) with the NNLO corrections, a had, NNLO VP The most recent update from DHMZ19 has a larger uncertainty compared to that of DHMZ17, since DHMZ19 have included an additional error to account for the difference they obtain for a π + π − µ when discarding either the KLOE or the BaBar data. As the KNT π + π − data combination benefits from stronger constraints imposed by the correlated uncertainties, the difference observed in a π + π − µ when discarding the data from either experiment is less severe. Therefore, and remembering also that data tensions are quantitatively accounted for in the resulting cross section by the local χ 2 error inflation, no additional uncertainty for aµ is applied in this analysis. 0.01) × 10 −10 [24], the total hadronic VP contribution to a µ is estimated to be a had, VP µ = (684.19 ± 2.38 tot ) × 10 −10 , (3.8) where, as in the case of the electron, the errors have been added linearly due to the full correlation between the R-ratio input for the three contributions. When considering the SM prediction, in the case of the muon (l = µ), the other contributions in equation (1.1) require reconsideration.
In contrast to the case of the electron, the muon is, at the current level of accuracy, not sensitive to the choice of either α(Rb) or α(Cs), or the updated five-loop QED contributions from [11]. Hence the value of the QED contributions, to the accuracy needed and quoted here, is unchanged at a QED µ = (11658471.90±0.01)×10 −10 [11,55]. For the EW contributions, the value chosen here is also the same as in [1]. However, it should be noted that an independent numerical evaluation of the two-loop EW contributions was recently performed [56], resulting in an estimate of the total EW contributions of a EW µ = (15.29 ± 0.10) × 10 −10 . This is consistent with the previously chosen value of a EW µ = (15.36 ± 0.10) × 10 −10 [57] and therefore no adjustment is made for this analysis.
For the hadronic LbL sector, in [1] the commonly quoted 'Glasgow consensus' estimate of a had, LbL µ ('Glasgow consensus') = (10.5 ± 2.6) × 10 −10 [58] was used, adjusted for a re-evaluation of the contribution to a had,LbL µ due to axial exchanges [59][60][61]. This led to a had, LbL µ = (9.8 ± 2.6) × 10 −10 [61] being adopted for the KNT18 analysis. Since that time, the progress in determining a had,LbL µ using dispersive approaches (where dispersion relations are formulated that allow for the determination of the hadronic LbL contributions from experimental data) has been significant. 7 These determinations are of particular interest for this analysis, as the fundamental approach to this work (and the works preceding it [1,22,39,40]) is that any estimates given be as model-independent and/or as data-driven as possible. With the contributions to the 'Glasgow consensus' estimate having been solely determined through model-dependent approaches, moving towards data-based evaluations of the hadronic LbL contributions is consistent with the general methodology of this undertaking.
Those hadronic LbL contributions that have been determined by dispersive techniques are the pseudoscalar poles (π 0 , η, η ) [62][63][64], the pion/kaon-box contributions [9,65] and the S-wave ππ rescattering contributions [65,66]. In addition, a new analysis of (longitudinal) short distance constraints has very recently become available [67,68], complementing the dispersive determination of the pseudoscalar contributions. The values for these contributions and their counterparts from the 'Glasgow consensus' estimate are shown in Table 3, where the estimate of the pseudoscalar contributions of the 'Glasgow consensus' already contains short distance contributions. With the aim to strive for a more model-independent approach, the value for a had, LbL µ in this work is taken as the sum of the contributions determined via dispersive approaches, the new estimates of short distance and charm quark corrections, plus the sum of the contributions from scalars, tensors and axial-vectors remaining from the original 'Glasgow consensus' estimate. 8 This results in a value for the total hadronic LbL contribution of a had, LbL µ = (9.34 ± 2.92) × 10 −10 , where the errors from the individual contributions have been summed linearly. This provides Contribution 'Glasgow consensus' [58] Dispersive evaluations   a conservative estimate of the overall uncertainty and also accounts for currently unavailable transverse short distance constraints, which are estimated to be sub-leading. The values for the contributions from all the individual sectors of the SM chosen in this analysis are summarised in Table 4. Summing these contributions together results in an updated SM prediction of the anomalous magnetic moment of the muon of a SM µ = (11 659 181.08 ± 3.78) × 10 −10 , where the uncertainty is determined from the uncertainties of the individual SM contributions added in quadrature. This value deviates from the current experimental measurement [5] by ∆a µ = (28.02 ± 7.37) × 10 −10 , (3.10) corresponding to a muon g − 2 discrepancy of 3.8σ. This result is compared with other determinations of a SM µ in Figure 9. The value for a SM µ in equation (3.9) has decreased by 0.96 × 10 −10 Figure 9: A comparison of recent and previous evaluations of a SM µ . The analyses listed in chronological order are: DHMZ10 [51], JS11 [52], HLMNT11 [40], FJ17 [53] and DHMZ17 [54], KNT18 [1] and DHMZ19 [37]. The prediction from this work is listed as KNT19, which defines the uncertainty band that other analyses are compared to. The current uncertainty on the experimental measurement [2][3][4][5] is given by the light blue band. The light grey band represents the hypothetical situation of the new experimental measurement at Fermilab yielding the same mean value for a exp µ as the BNL measurement, but achieving the projected four-fold improvement in its uncertainty [6].
compared to the KNT18 analysis [1]. This change comes, in nearly equal parts, from the reduction in the mean value of a had, LO VP µ and the new estimate of a had, LbL µ in this work. The increase in the uncertainty with respect to [1] comes from the increase in the error of a had, LbL µ owing to the changes in the estimate of this contribution discussed previously. Together, these have resulted in the increased discrepancy from 3.7σ in the KNT18 analysis to 3.8σ in this work.

The anomalous magnetic moment of the tau lepton, a τ
In the case of the τ , the determination of the LO hadronic VP contributions yields Note that in the case of the τ , the total NLO contributions are positive, while they are negative for the electron and muon, and any estimate based on a naive mass-scaling of the result for the muon would fail completely. The results for a had, LO VP τ from the individual hadronic channels are given in Table 1. Comparing with the evaluation in [19], which resulted in a had, LO   (337.5 ± 3.7) × 10 −8 , and a had, NLO VP τ = (7.6 ± 0.2) × 10 −8 obtained already in [23], there is consistency between the mean values found in the different analyses. However, there is a large reduction in the error in this work which is mainly due to the abundance of precise new data since [19]. Utilising the values from [19] for the QED, EW and hadronic LbL contributions (listed in Table 5), the updates to the hadronic VP contributions result in a SM prediction for the anomalous magnetic moment of the tau lepton of a SM τ = (117717.1 ± 3.9) × 10 −8 . (3.13) With the uncertainties of the hadronic VP contributions significantly improved, the uncertainty of a SM τ is now dominated by the hadronic LbL contributions, which account for ∼ 60% of the total error. However, it should be noted that the QED contributions, at ∼ 26% of the total error, are now less precise than the hadronic VP contributions. As explained in [19], the entire error δa QED τ ∼ 2 × 10 −8 is assigned as the uncertainty due to the missing contributions at fourloop (and beyond), and are crudely estimated from logarithmically enhanced terms expected at four-loop level. This indicates that a calculation of a QED τ at four loops would significantly improve the determination of a SM τ . Although, as stated in Section 1, the precision of the current experimental measurement of a exp τ = −0.018 (17) [18] makes a meaningful comparison between theory and experiment futile, this analysis confirms a difference ∆a τ = a exp τ − a SM τ at the level of 1σ as found in [18]. While at present there seems little prospect for an experiment dedicated to measuring a τ , it is not imperceivable to imagine that this might become possible in the future. Indeed, the additional potential for new physics discoveries due to the higher mass scale of the τ compared to the electron or the muon make this an interesting consideration.

Mu
For many years, precision measurements of the ground-state hyperfine splitting (HFS) of muonium ∆ν Mu served as a rigorous test of QED. Today, it still provides the best approach for determining the value of the electron-to-muon mass ratio and, therefore, the muon mass. As, like with the lepton g − 2, ∆ν Mu is sensitive to quantum effects, any differences in the comparison of experimental and theoretical determinations could be an indication of new physics. The current most precise experimental measurements of ∆ν Mu [75,76]  With the most recent of these measurements having been performed more than 20 years ago, the MuSEUM experiment at J-PARC is currently in the process of measuring the HFS of muonium (and the electron-to-muon mass ratio) with an aim to reduce the uncertainty in equation (3.17) by an order of magnitude [77].
Here, ν F denotes the so-called Fermi energy, where R ∞ is the Rydberg constant. The kernel function K Mu (s) is described in detail in [17]. Now utilising the compilation of the hadronic cross section determined in this work (see Section 2), the updated value for the hadronic VP contributions to the ground-state HFS of muonium are found to be ∆ν had, VP Mu = (232.04 ± 0.38 stat ± 0.66 sys ± 0.08 vp ± 0.27 fsr ) Hz = (232.04 ± 0.82 tot ) Hz . (3.22) Here, a noticeable mean value reduction and an uncertainty reduction of ∼ 43% compared to equation (3.19) are observed, which is in accordance with the same trends seen in the development of the corresponding determinations of a µ over the same period. Adjusting the theoretical prediction in equation (

Conclusions and future prospects
This analysis, KNT19, has presented updated evaluations of the hadronic vacuum polarisation contributions to the anomalous magnetic moment of the electron (a had, VP e ), muon (a had, VP µ ) and tau lepton (a had, VP τ ), to the ground-state hyperfine splitting of muonium (∆ν had, VP Mu ), and has also updated the value of the hadronic contributions to the running of the QED coupling at the scale of the mass of the Z boson (∆α had (M 2 Z )). These quantities are calculated using the hadronic R-ratio, obtained from a compilation of all available e + e − → hadrons cross section data. In this work, the data compilation has been updated from the determination in [1], accounting for new measurements. In the dominant π + π − channel, the inclusion of the CLEOc data [25] has increased the mean value slightly and marginally improved the uncertainty of a π + π − µ . In the π + π − π 0 channel, adjustments have been made to the treatment of the narrow ω resonance, which is now integrated over using a quintic polynomial interpolation in order to avoid an overestimation of the cross section from a linear interpolation that was recently noted in [38]. This has reduced the mean value of a π + π − π 0 µ by ∼ 1 × 10 −10 and, in turn, contributed to a significant reduction of the mean value of a SM µ in this work, although it is important to note that all estimates from this analysis are consistent with those given in [1]. In addition, other new measurements have been included which have removed the need to rely on isospin relations to estimate cross sections in three sub-leading channels, where in each case the new data agree well with the predictions of the KNT18 analysis.
The resulting hadronic R-ratio has been used as input into dispersion relations to determine a had, VP l (l = e, µ, τ ) at LO and NLO, ∆α had (M 2 Z ) and ∆ν had, VP

Mu
. This work has found ∆α (5) had (M 2 Z ) = (276.09 ± 1.12 tot ) × 10 −4 which has yielded a value for the QED coupling at the Z boson mass of α −1 (M 2 Z ) = 128.946 ± 0.015, which is consistent with [1]. For the hadronic VP contributions to the ground-state hyperfine splitting of muonium, the new data compilation gives ∆ν had, VP Mu = (232.04 ± 0.82 tot ) Hz, which is consistent with the previous determination of this quantity in [17], but constitutes a significant uncertainty reduction of ∼ 43%. A similar error reduction has been observed in the determination of the anomalous magnetic moment of the electron compared to [17], with this analysis finding a had, LO VP e = (186.08 ± 0.66 tot ) × 10 −14 . This, coupled with new estimates for the NLO contributions, translates to differences between experiment and theory of ∆a e (α Rb ) = (−1.312 ± 0.773) × 10 −12 (1.7σ) and ∆a e (α Cs ) = (−0.890 ± 0.362) × 10 −12 (2.5σ), depending on whether the QED contributions are determined using α measured via Rb or Cs atomic interferometry. For the muon g − 2, the new KNT19 analysis gives a had, LO VP µ = (692.78±2.42 tot )×10 −10 and a had, NLO VP µ = (−9.83±0.04 tot )×10 −10 . New choices in this work for the hadronic LbL contributions based on recent results from dispersive approaches (which have already significantly consolidated the 'Glasgow consensus'), coupled with the contributions from the other sectors of the SM, have resulted in a new estimate for the Standard Model prediction of a SM µ = (11 659 181.08 ± 3.78) × 10 −10 , which deviates from the current experimental measurement by 3.8σ. In the case of the τ , the value at LO is a had, LO VP τ = (332.81 ± 1.39 tot ) × 10 −8 , consistent with the value found in [19], but with an uncertainty that is smaller by ∼ 62%. Unfortunately, the current experimental bounds of the measured value of a τ are not stringent enough to draw any strong conclusions from the comparison between experiment and theory.
It is interesting to compare the values and uncertainties of a had, LO VP l and a SM l of the different leptons, which are shown in Table 7. Here, especially in the case of the hadronic contributions, the difference in the resulting magnitudes of these values due to lepton massscaling arguments is evident. Indeed, in the most extreme example, the value of a had, LO VP O(10 6 ) times larger for the τ than for the electron. For a SM l , the most striking difference is in the level of the precision between the different leptons. The electron, being less sensitive to hadronic effects than the muon or the τ , is by far the most precise. However, the larger uncertainty of a SM τ compared to a SM µ is not solely due to hadronic contributions (where, for the muon, the hadronic LbL estimates are more accurate than for the τ ). Instead, as noted in Section 3.3, the uncertainty assigned due to the missing four-loop contributions is a main cause of this disparity and could be improved through the calculation of a QED τ at four-loop order. With the tantalising prospect of new experimental measurements of a µ from Fermilab in the near future, and later from J-PARC, the predictions of a had, VP µ and a SM µ have been re-examined in detail and found to be robust. The opportunity to further improve the hadronic VP contributions estimated by dispersive approaches (as in this analysis) largely rests on new hadronic cross section measurements. For the π + π − channel, new measurements currently under analysis from the CMD-3, SND and BaBar experiments are eagerly awaited. Although these measurements are important in terms of improving the overall precision of a had, VP µ , it is hoped that they will help to resolve the lingering deviation between the KLOE [28][29][30][31] and BaBar [32] measurements, which drive the data tensions in a π + π − µ . In addition, expected data for the π + π − π 0 , π + π − π 0 π 0 and the inclusive channels, will be very beneficial. In preparation for the new experimental measurements of a µ , the efforts of the Muon g − 2 Theory Initiative [9] (and the groups within it) have already led to impressive achievements with regards to advancing the determinations of the hadronic VP and hadronic LbL contributions. Of great interest are the results from lattice QCD, which already provide first-principles cross checks of the now very precise data-driven estimates for the hadronic contributions to a SM µ . These are expected to become competitive with the current determinations within the next few years. Given the continued advancements in the theoretical predictions of a µ , coupled with the substantial progress of the experimental community, the study of the muon anomalous magnetic moment has never been better placed to severely constrain many scenarios for new physics beyond the SM, or, should the muon g − 2 discrepancy become fully established, to claim a discovery of new physics.