τ data-driven evaluation of Euclidean windows for the hadronic vacuum polarization

We compute for the first time the τ data-driven Euclidean windows for the hadronic vacuum polarization contribution to the muon g − 2. We show that τ -based results agree with the available lattice window evaluations and with the full result. On the intermediate window, where all lattice evaluations are rather precise and agree, τ -based results are compatible with them. This is particularly interesting, given that the disagreement of the e + e − data-driven result with the lattice values in this window is the main cause for their discrepancy, affecting the interpretation of the a µ measurement in terms of possible new physics.


Introduction
The first Fermilab measurement of the muon anomalous magnetic moment (a µ = (g µ − 2)/2, with g the gyromagnetic factor) confirmed [1] the final result from Brookhaven [2], yielding the world average a Exp µ × 10 11 = 116592061 (41) . ( This reaffirmed and strengthened the interest of the high-energy physics community in this observable, given its 4.2σ tension with the Standard Model prediction , obtained within the Muon g-2 Theory Initiative [38] 1 a SM µ × 10 11 = 116591810 (43) . (2) However, the BMWc lattice Collaboration published [66] a precise lattice-QCD evaluation of the hadronic vacuum polarization (HVP) contribution with the result a BMWc µ × 10 10 = 707.5 ± 5.5 , which yields an a µ evaluation at 1.5σ from a Exp µ , Eq. ( 1), and at 2.1σ from the data-driven result, based on σ(e + e − →hadrons), that was employed to get Eq.(2).A direct comparison between the BMWc and data-driven predictions for the HVP contribution can be found in Table 4.
3 Indeed it was during the refereeing of this article, [75]. 4This contribution yields a bit more than 70% of a HVP,LO µ .
was competitive with the e + e − data, although this is generally not considered the case at the moment [38].
Recently, a new τ data-driven approach was performed in Ref. [101].In this section, we utilize their results to evaluate the leading HVP contribution to the anomalous magnetic moment of the muon in the so-called window quantities [12,71].For this enterprise, we make use of the weight functions in center-of-mass energy Θ(s) from Eq. ( 12) in Ref. [69] which are related to those in Euclidean time [12] Θ The subscripts in Eq. ( 6) refer to the short-distance (SD), intermediate (win, although we will use int in the following) and long-distance (LD) contributions with parameters where and the IB corrections are encoded in The S EW term in Eq. ( 8) includes the short-distance electroweak corrections [113][114][115][116][117][118][119][120].FSR refers to the Final-State-Radiation corrections to the π + π − channel [121,122] 7 , while the G EM (s) factor includes the QED corrections to the τ − → π − π 0 ν τ decay with virtual plus real photon radiation.
a phase space correction owing to the π ± − π 0 mass difference.The last term in R IB (s) corresponds to the ratio between the neutral (F V (s)) and the charged (f + (s)) pion form factor, which includes one of the leading IB effects, the ρ 0 − ω mixing correction.
The IB corrections to a HVP, LO µ using τ data in the dominant ππ channel [123,124] can be evaluated using the following expression [98] ∆a which measures the difference between the correct expression for σ 0 ππ(γ) and the naive Conserved Vector Current approximation, with S EW = 1 and R IB (s) = 1.
These contributions are summarized in Table 1 for each IB correction.
• G EM (s) was originally computed in Ref. [97] including those operators yielding resonance saturation of the O(p 4 ) chiral couplings in the frame of Resonance Chiral Theory (RχT) [127-130] 9 10 , which comprises Goldstone bosons and resonance fields extending the χPT framework [134][135][136] to higher energies.A recalculation of G EM (s) was performed later in [137,138] using a Vector Meson Dominance (VMD) model [139].In Ref. [101] (see also [140]), two of us explored the impact of the RχT operators contributing to resonance saturation at the next chiral order (O(p 6 )) on the G EM (s), as well as estimating the uncertainty of the original computation in [97].These results [101] are consistent with the earlier RχT and the VMD estimates [101].Availing of these results, G EM (s) produces a correction of ∼ −3.5%, ∼ −17.1% and −17.1 ) for the SD, int and LD contributions, respectively.Interestingly, the SD and int contributions at O(p 4 ) change in sign while this is not the case at O(p 6 ) where all the contributions are always negative.
• The ratio of the form factors (FF) gives an overall correction of ∆a HVP, LO µ = +7.13(1.48)( 1. 59  1.54 )( 0.85 0.80 )• 10 −10 , from which ∼ 2.2%, ∼ 26.8% and ∼ 71.0% stand for the SD, int and LD corrections, respectively.The errors quoted in this contribution correspond to the electromagnetic shifts in the widths and masses of the ρ meson, and to the ρ 0 − ω mixing parameter [97,141] (see Eqs. (5.5) and (5.6) in Ref. [97]), respectively 11 .For this analysis, we use the same numerical inputs as in [101].The central value reported in Table 1 corresponds to the weighted average between the FF1 and FF2 sets 12 .
The overall correction is also consistent with those in Refs.[97,98,101].) for 2π below 1.0 GeV using the parameters in Eq. (7).The blue region corresponds to the experimental average from τ data.The e + e − number was taken from Ref. [69].Using the τ spectral functions measured by ALEPH [143], Belle [144], CLEO [145] and OPAL [146], we evaluate a HVP, LO µ [ππ] using the window parameters in Eq. ( 7).These results are outlined in Tables 2 and 3 for s ≤ 1, 2, 3, and 3.125 GeV 2 , i.e., integrating Eq. ( 4) with σ 0 ππ(γ) in Eq. ( 8) from s thr = 4m 2 π up to some given cut-off.In the aforementioned tables 2 and 3, the first uncertainty is connected to the systematic errors on the mass spectrum, and from the τ -mass and V ud uncertainties; the second error is associated to B ππ 0 and B e ; and the third one is due to the IB corrections.The Mean value in the tables corresponds to the weighted average from the different window contributions for each experiment, the first error is related to the experimental measurements, while the second one comes from the IB corrections.
An evaluation of a HVP µ in the windows in Euclidean time using e + e − data was performed in Ref. [69] using the parameters in Eq. (7), see Table 4 below.A comparison between these window quantities for HVP in the 2π channel below 1 GeV amounts to a discrepancy of 4.0 σ, 2.9 σ and 1.8 σ between the τ and e + e − evaluations applying the G EM (s) correction at O(p 4 ) in RχT to the τ data for the SD, int and LD contributions, respectively.On the other hand, when we include the corrections at O(p 6 ), the difference between these two evaluations decreases to 3.3 σ, 2.3 σ and 0.7 σ for the SD, int and LD contributions, respectively.These results are depicted in Figs. 1 and 2, where the blue band corresponds to the experimental average using τ data.Fig. 3 shows a zoomed comparison between τ (after IB corrections at O(p 6 )) and e + e − → π + π − spectral function using the ISR measurements from BABAR [88] and KLOE [86] (left-hand panel) and the energy-scan measurements from CMD-3 [76] (right-hand panel).Colored bands correspond to the weighted average of the uncertainties coming from both sets of data in each figure.Although it may seem obvious than increased IB-corrections in the ρ region will increase the compatibility between τ and CMD-3 e + e − data, dedicated studies (like e.g.Ref. [147]) seem necessary to fully understand this issue (even more in the comparison with KLOE and BaBar).
A direct comparison between a HVP, LO µ [ππ, τ ] and the lattice results is not possible.For that endeavour, it is necessary to supplement the 2π evaluation with the remaining contributions from all other channels accounting for the hadronic cross-section.To illustrate the impact of this con- [ππ, τ ] in units of 10 −10 at O(p 4 ) in RχT using the experimental measurements from the ALEPH [143], Belle [144], CLEO [145] and OPAL [146] Colls.The first error is related to the systematic uncertainties on the mass spectrum and also includes contributions from the τ -mass and V ud uncertainties.The second error arises from B ππ 0 and B e , and the third error comes from the isospin-breaking corrections.The uncertainties in the mean value correspond to the experiment and to the IB corrections, respectively.
tribution in a HVP, LO µ , we follow two approaches 13 .Firstly, using the values reported in Table 1 of Ref. [69] we subtract the contribution from the 2π channel below 1.0 GeV (we represent this procedure with '< 1 GeV') and replace it by the corresponding mean value in Tables 2 and 3.This way, we get Secondly, we estimate the 2π full contribution using the ratios a I µ /a HVP, LO µ , where I stands for SD, int and LD, from the corresponding window quantities in Ref. [69] and the overall weightedaverage evaluation of a HVP, LO µ [ππ, e + e − ] = 505.1(1.7)× 10 −10 in Refs.[7,8].However, as can be easily computed from Tables 2 and 3, these ratios are not the same between the second and the last column.This effect is mainly due to the weight functions Θ(s) in Eq. ( 6) 14 , so, in order to take this into account, we use the difference between the ratio from the second and third column to correct the ratios from e + e − data.Then we subtract this value from the total contribution and replace it by our results 15 .Finally, we get at O(p 6 ).All these results are reasonably consistent with each other.We summarize these outcomes in Table 4 along with the lattice results [12,62,[66][67][68]148] and other e + e − data-driven evaluations [38,69].These numbers are depicted in Fig. 4 for the intermediate window, where the blue band represents the weighted average of the lattice results, a int µ = 235.8(6)• 10 −10 , excluding those from RBC/UKQCD 2018 [12] and ETMC 2021 [148] collaborations.The contributions of the intermediate window using τ data are slightly closer to the results from lattice QCD than to the e + e − values.Therefore, the ∼ 4.3σ discrepancy between the e + e − data-driven and lattice evaluations is reduced to ∼ 1.5σ when τ data is used for the 2π channel.On the other hand, there is only one lattice result for the short-distance window [68] which seems to be in agreement with both data-driven HVP evaluations.in units of 10 −10 .The first two rows correspond to the τ evaluation in the first approach, while rows 3 and 4 are the evaluations in the second one.The rows 5-10 are the lattice results [12,62,[66][67][68]148].The last rows are the evaluations obtained using e + e − [4,38,66,69,147] and τ data [125].See Fig. 11 in Ref. [62] for more details.

Conclusions
While the BNL and FNAL measurements of a µ agree nicely within errors, the situation is not that clear for its SM prediction's counterpart.On the one hand, data-driven methods based on e + e − →hadrons data have to deal with the tensions between experiments (particularly among BaBar and KLOE, and now with CMD-3), which makes the computation of the uncertainty in Eq. ( 2) a non-trivial task, as will be its update.On the other side, there is still only one lattice QCD evaluation (BMWc Coll.) of a HVP µ with competitive uncertainties, that lies between a Exp µ and its SM data-driven prediction.However, the most recent Mainz/CLS, ETMC, and RBC/UKQCD results have similar errors to BMWc in the intermediate window, where all of them agree remarkably.It is long-known that alternative data-driven evaluations are possible, utilizing this time semileptonic τ -decay data (and isospin-breaking corrections, with attached model-dependent uncertainty), as we have done here.
In this context, we have applied the study from Ref. [62], that computed window quantities in Euclidean time for data-driven evaluations of a HVP µ using e + e − → hadrons data, to the semileptonic τ decays case (focusing on the dominant two-pion contribution).Our main results are collected in Table 4 and show that τ -based results are compatible with the lattice evaluations in the intermediate  .The blue band corresponds to the weighted average of the lattice results excluding RBC/UKQCD 2018 [12] and ETMC 2021 [148].
window, being the e + e − -based values in tension with both of them.This difference is the main cause for the larger discrepancy of the latter with a Exp µ and should be further scrutinized.Supplemented by the relevant IB corrections computed on the lattice [106], the results in this work could be used together with lattice QCD to obtain an alternative data-based determination from τ decays (and lattice QCD) which can be helpful in solving the present puzzle.

Figure 1 :
Figure 1: Windows quantities for HVP at O(p 4) for 2π below 1.0 GeV using the parameters in Eq.(7).The blue region corresponds to the experimental average from τ data.The e + e − number was taken from Ref.[69].

Figure 4 :
Figure 4: Comparison of different evaluations of the total intermediate window contribution to a HVP, LO µ which correspond to inverse energies of the order of 500, 200 and 1300 MeV, respectively.In what follows, we will focus on the dominant 2π contribution only.Including isospin breaking (IB) corrections, i.e., O[(m u − m d )p 2 ] and O(e 2 p 2 ) contributions, the bare hadronic e + e [96,97]section σ 0 ππ(γ) is related to the observed differential τ decay rate dΓ ππ[γ] through[96,97]