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Window quantities for the hadronic vacuum polarization contributions to the muon anomalous magnetic moment in spacelike and timelike domains

A. V. Nesterenko

Abstract

The relations between the window quantities for the hadronic vacuum polarization contributions to the muon anomalous magnetic moment $a^{\text{HVP}}_μ$ in spacelike and timelike domains are studied. Two types of window functions (abrupt and smooth) as well as two kinds of kinematic intervals (symmetric and asymmetric with respect to the spacelike/timelike flip) are addressed. It is shown that the window quantities for $a^{\text{HVP}}_μ$ represented in terms of the hadronic vacuum polarization function $\barΠ(Q^2)$, the Adler function $D(Q^2)$, and the $R$-ratio of electron-positron annihilation into hadrons are mutually equivalent only if the additional contributions due to the window edge effects are properly taken into account and the explicit expressions for such contributions are derived. The obtained results enable one to evaluate $a^{\text{HVP}}_μ$ by making simultaneous use of the inputs for functions $\barΠ(Q^2)$, $D(Q^2)$, and $R(s)$ at various energies and an example of such hybrid assessment is provided. The obtained results also enable one to accurately compare the window quantities for $a^{\text{HVP}}_μ$ based, e.g., on MUonE or lattice data with the ones based on $R$-ratio data, even if the window function covers different kinematic ranges in spacelike and timelike domains.

Window quantities for the hadronic vacuum polarization contributions to the muon anomalous magnetic moment in spacelike and timelike domains

Abstract

The relations between the window quantities for the hadronic vacuum polarization contributions to the muon anomalous magnetic moment in spacelike and timelike domains are studied. Two types of window functions (abrupt and smooth) as well as two kinds of kinematic intervals (symmetric and asymmetric with respect to the spacelike/timelike flip) are addressed. It is shown that the window quantities for represented in terms of the hadronic vacuum polarization function , the Adler function , and the -ratio of electron-positron annihilation into hadrons are mutually equivalent only if the additional contributions due to the window edge effects are properly taken into account and the explicit expressions for such contributions are derived. The obtained results enable one to evaluate by making simultaneous use of the inputs for functions , , and at various energies and an example of such hybrid assessment is provided. The obtained results also enable one to accurately compare the window quantities for based, e.g., on MUonE or lattice data with the ones based on -ratio data, even if the window function covers different kinematic ranges in spacelike and timelike domains.
Paper Structure (12 sections, 74 equations, 6 figures, 1 table)

This paper contains 12 sections, 74 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Results
  3. Constant window function
  4. Abrupt symmetric window function
  5. Abrupt asymmetric window function
  6. Smooth symmetric window function
  7. Smooth asymmetric window function
  8. Discussion
  9. General comments
  10. Hybrid evaluation of $a^{\text{\normalfont{HVP}}}_{\mu}$
  11. Updated DPT value of $a_{\mu}$
  12. Conclusions

Figures (6)

  • Figure 1: Left--hand plot: the constant window function $W_{1}(q^2)$ (\ref{['DefW1']}). Right--hand plot: the closed integration contour $C$ in the complex $q^2$--plane in Eq. (\ref{['W1IntC1']}).
  • Figure 2: Left--hand plot: the abrupt symmetric window function $W_{2}(q^2)$ (\ref{['DefW2']}). Right--hand plot: the closed integration contours in the complex $q^2$--plane in Eq. (\ref{['IntC1C2']}).
  • Figure 3: Left--hand plot: the abrupt asymmetric window function $W_{3}(q^2)$ (\ref{['DefW3']}). Right--hand plot: the closed integration contours in the complex $q^2$--plane in Eq. (\ref{['IntC1C2W3']}).
  • Figure 4: Left--hand plot: the smooth symmetric window function $W_{4}(q^2)$ (\ref{['DefW4']}). Right--hand plot: the closed integration contour in the complex $q^2$--plane in Eq. (\ref{['IntW4']}).
  • Figure 5: Left--hand plot: the smooth asymmetric window function $W_{5}(q^2)$ (\ref{['DefW5']}). Right--hand plot: the closed integration contour in the complex $q^2$--plane in Eq. (\ref{['IntW5']}).
  • ...and 1 more figures