^{*}

^{†}

^{3}.

There are emerging tensions for theory results of the hadronic vacuum polarization contribution to the muon anomalous magnetic moment both within recent lattice QCD calculations and between some lattice QCD calculations and R-ratio results. In this paper, we work toward scrutinizing critical aspects of these calculations. We focus in particular on a precise calculation of Euclidean position-space windows defined by RBC/UKQCD that are ideal quantities for cross-checks within the lattice community and with R-ratio results. We perform a lattice QCD calculation using physical up, down, strange, and charm sea quark gauge ensembles generated in the staggered formalism by the MILC Collaboration. We study the continuum limit using inverse lattice spacings from

The established theory result for the muon anomalous magnetic moment,

For the HLbL contribution, new analytic approaches

For the HVP contribution, however, tensions exist within lattice QCD calculations

Concretely, there are two tensions for the isospin-symmetric quark-connected light-quark contribution (Fig.

To this end, we use the same lattice QCD ensembles as Aubin

Within our numerical framework, we can also access the connected strong-isospin-breaking contribution as well as the strange-quark connected contribution. We provide results also for these contributions including a wide range of different windows. We hope that these results will prove useful to further understand the current tensions.

This manuscript is organized as follows: Sec.

In this work, we perform a calculation of the HVP contribution to

Feynman diagrams for the isospin-symmetric contribution to the HVP. The dots represent the vector currents coupling to external photons. These diagrams represent gluon contributions to all orders.

Connected and disconnected strong-isospin-breaking (SIB) diagrams. The cross denotes the insertion of a scalar operator. Also here each diagram represents gluon contributions to all orders.

The diagrams

The weighting kernel in Eq.

It is instructive to isolate specific ranges of Euclidean time in order to better understand their contributions to

When performing computations with staggered quarks, parity projections are not possible and correlation functions receive contributions from parity partner states. These parity partners have different spin and taste quantum numbers and constitute unwanted contributions to the correlation function. The unwanted contributions come as oscillating terms with a prefactor proportional to

The computation in this work is performed with the highly improved staggered quark action for both valence and sea quarks. The ensembles were generated by the MILC Collaboration

List of ensemble parameters for ensembles used in this study. Table values reproduced from Table I of Ref.

Sources are inverted on random noise vectors that solve the Green’s function equation,

Results are computed on three ensembles with

Sea and valence quark masses used for each ensemble. Sea quark masses are listed in Ref.

Number of configurations and time sources used for each ensemble and valence quark mass combination. The valence quark masses are quoted as ratios of the valence quark mass to the tuned strange-quark mass,

For the two lightest masses

We show

The first part of the analysis consists of taking the continuum limit of each individual mass point. This extrapolation is applied before considering extrapolations in quark mass or volume.

Figures

Continuum extrapolation of the

Same as Fig.

Figures

Windows with

Same as Fig.

Figure

Continuum extrapolations with

Figure

Unimproved versus parity improved data as a function of lattice spacing. The upper-left, upper-right, lower-left, and lower-right plots have

After the continuum extrapolation, the valence quark masses are extrapolated to the isospin-symmetric light-quark mass. The value obtained from this extrapolation gives the connected light-quark contribution to the HVP from Eq.

Figure

Valence mass extrapolation of

List of fit parameters for the valence mass extrapolation in each window. The fits are parametrized as degree

Figure

Valence mass extrapolation for four different windows. The points are the continuum-extrapolated data and the shaded region is the mass extrapolation. For short-distance windows, a linear extrapolation is sufficient, while a quadratic fit is needed for long-distance windows. The fit parameters for each window are given in Table

Table

Results for

The finite-volume correction (FVC) is a correction to the long-distance physics in the correlation function due to the finite spatial extent of the lattice. The lattice states in the long-distance region are mostly composed of two-pion scattering states with zero center-of-mass momentum, up to mixing with other states that share the same quantum numbers. The finite spatial extent imposes a lower limit on the size of a unit of momentum,

The estimate of the FVC is obtained from the Lellouch-Lüscher-Gounaris-Sakurai procedure

The

These numbers are listed in Table

Finite-volume corrections for each window. The numbers for the light-quark connected isospin-symmetric contribution are plotted in the bottom panel of Fig.

The size of the contribution from each light and strange window and the size of the finite-volume correction for the light-quark mass for each of the

The top panel shows the continuum, infinite-volume limit of the connected isospin-symmetric light and strange windows with

Final results, including finite-volume corrections, for connected isospin-symmetric light and strange-quark contributions.

Since at leading order in

In order to address this issue, we use NLO PQChPT

It is instructive to consider the infinite-volume NLO PQChPT results,

We summarize our results for the total contributions to

Overview of results for

Overview of total results for

Overview of individual contributions to

We combine new results obtained in this paper with results for the missing contributions from RBC/UKQCD

We provide results for the connected SIB contribution both at finite volume (

We hope that the larger set of window results provided in this work can be useful to further scrutinize the emerging tensions within the lattice QCD community and within lattice QCD and the R ratio. We expect that in the near future other lattice collaborations will also provide results for the total

We thank our colleagues in the RBC and UKQCD Collaborations for interesting discussions. We thank the MILC Collaboration for the ensembles used in this analysis. Inversions and contractions were performed with the MILC code version 7. This work was supported by resources provided by the Scientific Data and Computing Center (SDCC) at Brookhaven National Laboratory (BNL), a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy. The SDCC is a major component of the Computational Science Initiative at BNL. We gratefully acknowledge computing resources provided through USQCD clusters at BNL and Jefferson Lab. C. L. and A. S. M. are supported in part by U.S. DOE Award No. DESC0012704(BNL) and by a DOE Office of Science Early Career Award.

This section contains tables with a detailed breakdown of systematic errors from the window data that appear in Table

A detailed breakdown of the systematic uncertainties for the data in the column labeled

Same as Table

Same as Table

Same as Table

Same as Table

Same as Table