^{3}.

The ratios among the leading-order (LO) hadronic vacuum polarization (HVP) contributions to the anomalous magnetic moments of an electron, muon, and

Since many years a long-standing deviation between experiment and theory persists for the anomalous magnetic moment of the muon,

On the theoretical side the present accuracy of the Standard Model (SM) prediction is at a similar level, 0.53 ppm

A new interesting deviation occurs in the case of the anomalous magnetic moment of the electron

On the contrary no direct measurement of the anomalous magnetic moment of the third charged lepton of the SM, the

In this respect note that the absolute value of the electron anomaly

For the three leptons the SM prediction of their anomalous magnetic moments is given by the sum of three contributions

Precise determinations of

During the last years a tremendous effort has been put to obtain accurate determinations of both

As far as the electron and the

The aim of this work is to present a lattice determination of the ratios of the leading-order (LO) HVP contributions to the lepton anomalous magnetic moments

Our simulations include the effects at order

We stress that the hadronic quantities

For the electron-muon ratio we get

In Eq.

Our result

Let us now introduce the following HVP quantities

The results

The plan of the paper is as follows.

In Sec.

In Sec.

In Sec.

Section

The LO HVP contribution

The HVP form factor

In this work we adopt the time-momentum representation of Ref.

The LO HVP contributions

Comparison of the normalized quantities

It can be seen that the time distances relevant for the integration in the rhs of Eq.

Thanks to recent progress in lattice

It should be stressed that the separation given in Eq.

Since all quark flavors contribute to the em current

In the case of the muon the bottom-quark LO HVP contribution

Correspondingly, from Eqs.

We start by considering the electron-muon ratio

In the next two subsections we address separately the determination of

The results obtained for the ratio

Results for the (connected) light-quark contribution to the electron-muon ratio,

A few comments are in order.

The precision of the data ranges from

The uncertainties of the data are mainly related to the statistical errors and to a lesser extent to the scale setting.

By switching off the uncertainties of the scale setting in the bootstrap samples of Appendix

Finite-volume effects (FVEs) are clearly visible in the case of the four gauge ensembles A40.XX (see Appendix

The pion mass dependence is significative and the extrapolation to the physical pion mass requires a careful treatment, while discretization effects appear to be subleading.

In order to remove FVEs from the data we follow the approach of Ref.

Using the analytic representation, the FVEs on

The FVEs are subtracted from the data at finite volume using the following formula:

The final steps are the extrapolations to the physical pion mass and to the continuum limit. For evaluating the former one, which represents the dominant source of the systematic uncertainty, we adopt three strategies, which will be described in what follows.

In Ref.

Therefore, we adopt the following ansatz:

In Eq.

The results obtained with the fitting function

Values of the ratio

At the physical pion mass and in the continuum limit the first strategy yields

However, the chiral enhancement of

According to Sec. II of Ref.

As suggested by the smallness of the discretization effects exhibited by the data in Fig.

Finally, we adopt a third strategy based on the use of the analytic representation of the vector correlator

As shown in Ref.

The first five moments of the polarization function have been evaluated using the

Courtesy of A. Keshavarzi, D. Nomura and T. Teubner.

The corresponding predictions based on our analytic representation at the physical point were found to be nicely consistent with the above dataOur analytic representation includes not only the contribution of the isospin-1

Thus, besides the statistical uncertainties of the parameters appearing in the representation, the only important source of the systematic error comes from their chiral extrapolation. For estimating the corresponding systematics we have tried several fitting functions and we have checked that it suffices to consider the four different fits in which (i) the physical value of

Within the third strategy we get

By including in the systematic error the spread among the results of the three strategies [evaluated according to Eq. (28) of Ref.

In this section we determine the ratio

We make use of a simple procedure based on the values of

The values of the ratios

Values of the ratios

We can now evaluate the ratio

New recent estimates of both

Collecting our findings

Our result

We can apply the procedure described in Sec.

In Fig.

Comparison of the electron-muon ratio

Before closing this section we provide our results also for the electron-

The dependencies of

Values of the ratios

Our findings for both

In this work we have evaluated the ratios among the leading-order hadronic vacuum polarization contributions to the anomalous magnetic moments of electron, muon and

We have shown that in the case of the electron-muon ratio the hadronic uncertainties in the numerator and in the denominator largely cancel out, while in the cases of the electron-

We stress that the reduced sensitivity of

Using the present determinations of the muon

We gratefully acknowledge C. Lehner and M. Hoferichter for useful comments and V. Lubicz for a careful reading of the manuscript. We warmly thank F. Sanfilippo for providing us the code for calculating the light-quark contribution to the vector current-current correlator in the case of the ETMC gauge ensemble cB211.072.64

The gauge ensembles used in this work are those generated by ETMC with

We have performed simulations at three values of the inverse bare lattice coupling

Values of the valence and sea bare quark masses (in lattice units), of the pion, kaon and D-meson masses for the

At each lattice spacing, different values of the light sea-quark masses have been considered. The light valence and sea-quark masses are always taken to be degenerate. The bare masses of the valence strange and charm quarks (

In this work, as well as in all our works on the muon HVP terms

the continuum extrapolation adopting for the matching of the lattice scale either the Sommer parameter

the chiral extrapolation performed with fitting functions chosen to be either a polynomial expansion or a ChPT ansatz in the light-quark mass;

the choice between methods M1 and M2, which differ by

Statistical errors on the meson masses and the various HVP terms are evaluated using the jackknife procedure. The uncertainties based on data obtained from independent ensembles of gauge configurations, like the errors of the fitting procedures, are evaluated using the above bootstrap events in order to take properly into account cross-correlations. The results corresponding to the eight branches of the analysis are then averaged according to Eq. (28) of Ref.

The statistical accuracy of the meson correlator is based on the use of the so-called “one-end” stochastic method

In Table

Values of the (connected) light-quark contribution to the electron-muon ratio [see Eq.

For the second strategy adopted in Sec.

Following Ref.

The correlator

Since the effective threshold

The dual correlator

As is well known after Refs.

The two-pion contribution

The GS form factor

More precisely, for each lattice spacing and volume the following dimensionless parameters,

We want to highlight an important feature of the representation