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Low energy moments of heavy quark current correlators at four loops

A. Maier, P. Maierhoefer, P. Marquard, A. V. Smirnov

TL;DR

The paper addresses the challenge of extracting precise heavy-quark parameters by computing four-loop low-energy moments of heavy-quark current correlators. It introduces a novel reduction approach for integrals with internal self-energies, combining a two-step IBP procedure with region-bases in FIRE to manage self-energy insertions, and applies this to obtain the second and third moments for vector, axial-vector, and scalar currents, as well as the fourth moment for the pseudoscalar current via a Ward identity. The work provides full analytic and numerical four-loop contributions, confirms consistency with prior results, and yields updated charm- and bottom-quark masses and strong coupling values, reducing theoretical uncertainties. These advances improve the reliability of mass determinations from lattice and experimental data and enhance the reconstruction of the polarization function $\Pi(q^2)$ across energy ranges.

Abstract

We describe several techniques for the calculation of multi-loop integrals and their application to heavy quark current correlators. As new results, we present the four-loop correction to the second and third physical moment in the low-energy expansions of vector, axial-vector and scalar quark current correlators. Using a Ward identity, we obtain the third and fourth moment for the pseudo-scalar correlator. We briefly discuss the impact of these results on the determination of the charm quark mass and the strong coupling constant using lattice simulations for the current correlators and of the charm- and bottom-quark mass from experimental data for σ(e^+ e^- -> hadrons).

Low energy moments of heavy quark current correlators at four loops

TL;DR

The paper addresses the challenge of extracting precise heavy-quark parameters by computing four-loop low-energy moments of heavy-quark current correlators. It introduces a novel reduction approach for integrals with internal self-energies, combining a two-step IBP procedure with region-bases in FIRE to manage self-energy insertions, and applies this to obtain the second and third moments for vector, axial-vector, and scalar currents, as well as the fourth moment for the pseudoscalar current via a Ward identity. The work provides full analytic and numerical four-loop contributions, confirms consistency with prior results, and yields updated charm- and bottom-quark masses and strong coupling values, reducing theoretical uncertainties. These advances improve the reliability of mass determinations from lattice and experimental data and enhance the reconstruction of the polarization function across energy ranges.

Abstract

We describe several techniques for the calculation of multi-loop integrals and their application to heavy quark current correlators. As new results, we present the four-loop correction to the second and third physical moment in the low-energy expansions of vector, axial-vector and scalar quark current correlators. Using a Ward identity, we obtain the third and fourth moment for the pseudo-scalar correlator. We briefly discuss the impact of these results on the determination of the charm quark mass and the strong coupling constant using lattice simulations for the current correlators and of the charm- and bottom-quark mass from experimental data for σ(e^+ e^- -> hadrons).

Paper Structure

This paper contains 21 sections, 34 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Methods
  3. Reduction technique for integrals with internal self energies
  4. Algorithm
  5. Implementation in FIRE
  6. Top level reduction
  7. Results
  8. Applications
  9. Conclusion
  10. Analytical Results
  11. Second moments
  12. Vector current
  13. Scalar current
  14. Axial-vector current
  15. Third moments
  16. ...and 6 more sections

Figures (2)

  • Figure 1: The figure above shows an example of eq. (\ref{['T-to-se-topo']}) for a four-loop tadpole. Note that on the right hand side besides massless (dashed) lines and lines with mass $m$ (solid) also a line with mass $2m$ (double line) appears which is not present in the initial integral. The next step would be to repeat the procedure for the two-loop self energy.
  • Figure 2: All four-loop tadpoles with self energy insertions which appear in our calculation can be mapped to these seven topologies. Dotted lines are massless and solid lines carry the mass $m$. The wavy propagator which connects the self energies appears with the masses $km$ where $k$ is an element of the list below the corresponding diagram.