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Energy levels of multiscale bound states from QED energy-momentum trace

Michael I. Eides, Vladimir A. Yerokhin

TL;DR

This work shows that the bound-state energy levels in QED with multiple independent mass scales can be obtained from the matrix elements of the energy-momentum tensor (EMT) trace. By decomposing the one-loop EMT trace in muonic hydrogen into electron and muon mass terms plus the $F^2$ contribution, and by analyzing multiscale diagrams in the Furry picture, the authors demonstrate that EMT-trace diagrams arise as logarithmic derivatives with respect to all active masses of the standard energy-diagram sums. Explicit one-loop calculations for muonic hydrogen confirm that the EMT-trace contributions reproduce the conventional Lamb-shift corrections, including the electron-vacuum-polarization effects, and reveal cancellations governed by the beta-function structure. The results generalize the single-scale insight to multiscale bound states, providing a diagrammatic, mass-derivative-based link between EMT traces and bound-state energies with potential extensions beyond one loop. This offers a new perspective on precision QED in multiscale systems and could illuminate EMT-related aspects of bound-state physics in gauge theories.

Abstract

Energy levels of QED bound states, which depend on a number of independent mass parameters, can be calculated as matrix elements of the QED energy-momentum trace. As an example of such system we consider muonic hydrogen. The leading one-loop corrections to its energy levels depend on the electron and muon masses. These corrections are calculated as matrix elements of the energy-momentum trace. Respective one-loop trace diagrams are different from the standard Lamb shift diagrams. We explain analytically and diagrammatically why two different sets of diagrams lead to the same results. Similar relationships should also hold beyond the one-loop approximation.

Energy levels of multiscale bound states from QED energy-momentum trace

TL;DR

This work shows that the bound-state energy levels in QED with multiple independent mass scales can be obtained from the matrix elements of the energy-momentum tensor (EMT) trace. By decomposing the one-loop EMT trace in muonic hydrogen into electron and muon mass terms plus the contribution, and by analyzing multiscale diagrams in the Furry picture, the authors demonstrate that EMT-trace diagrams arise as logarithmic derivatives with respect to all active masses of the standard energy-diagram sums. Explicit one-loop calculations for muonic hydrogen confirm that the EMT-trace contributions reproduce the conventional Lamb-shift corrections, including the electron-vacuum-polarization effects, and reveal cancellations governed by the beta-function structure. The results generalize the single-scale insight to multiscale bound states, providing a diagrammatic, mass-derivative-based link between EMT traces and bound-state energies with potential extensions beyond one loop. This offers a new perspective on precision QED in multiscale systems and could illuminate EMT-related aspects of bound-state physics in gauge theories.

Abstract

Energy levels of QED bound states, which depend on a number of independent mass parameters, can be calculated as matrix elements of the QED energy-momentum trace. As an example of such system we consider muonic hydrogen. The leading one-loop corrections to its energy levels depend on the electron and muon masses. These corrections are calculated as matrix elements of the energy-momentum trace. Respective one-loop trace diagrams are different from the standard Lamb shift diagrams. We explain analytically and diagrammatically why two different sets of diagrams lead to the same results. Similar relationships should also hold beyond the one-loop approximation.
Paper Structure (12 sections, 40 equations, 6 figures)

This paper contains 12 sections, 40 equations, 6 figures.

Table of Contents

  1. Introduction
  2. One-loop EMT trace diagrams
  3. Multiscale Problems
  4. Diagrams with electron polarization
  5. One-loop electron polarization loop contribution to the Lamb shift
  6. Calculations of the trace diagrams in Fig. \ref{['vacpolmisel']}
  7. Matrix element of $m_\mu\bar{\psi}_\mu\psi_\mu$ with sidewise insertion of the polarization loop
  8. Matrix element of the scalar vertex $m_e\bar{\psi}_e\psi_e$ insertion in the polarization loop
  9. Matrix element of the anomalous term $(\beta_e/2e)F^2$ insertion in the Coulomb photon
  10. Summary
  11. One-loop renormalization constants
  12. Schrödinger-Coulomb bound state wave functions

Figures (6)

  • Figure 1: Self-energy type trace Lamb shift diagrams.
  • Figure 2: Muon vacuum polarization type trace Lamb shift diagrams.
  • Figure 3: Electron vacuum polarization type trace Lamb shift diagrams.
  • Figure 4: Classical electron polarization Lamb shift diagrams in muonic hydrogen.
  • Figure 5: Classical one-loop Lamb shift diagrams in electronic hydrogen.
  • ...and 1 more figures