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The ultrafine splitting of heavy quarkonium with next-to-next-to-next-to-next-to-leading-order accuracy

Jose M. Escario, Andreas Maier, Clara Peset, Antonio Pineda

Abstract

We compute the hyperfine splitting of P-wave heavy quarkonium states with next-to-next-to-next-to-next-to-leading-order accuracy. The resummation of logarithms with next-to-next-to-next-to-next-to-leading-logarithmic accuracy is also addressed. A phenomenological analysis of these results is performed for bottomonium, charmonium and the $B_c$ system. We also apply these results to positronium, muonium, hydrogen and muonic hydrogen.

The ultrafine splitting of heavy quarkonium with next-to-next-to-next-to-next-to-leading-order accuracy

Abstract

We compute the hyperfine splitting of P-wave heavy quarkonium states with next-to-next-to-next-to-next-to-leading-order accuracy. The resummation of logarithms with next-to-next-to-next-to-next-to-leading-logarithmic accuracy is also addressed. A phenomenological analysis of these results is performed for bottomonium, charmonium and the system. We also apply these results to positronium, muonium, hydrogen and muonic hydrogen.
Paper Structure (25 sections, 81 equations, 4 figures)

This paper contains 25 sections, 81 equations, 4 figures.

Table of Contents

  1. Introduction
  2. pNRQCD Lagrangian
  3. Potentials
  4. Energy-dependent terms and associated matching-scheme dependence
  5. ${\cal O}(\alpha_{\rm s}^3/m^2)$ spin-dependent and velocity-independent potential with Wilson loops
  6. One-loop potentials of order $\alpha_{\rm s}^2/m^3$
  7. Tree-level potentials of order $\alpha_{\rm s}/m^4$
  8. Quantum-mechanical perturbation theory
  9. First-order perturbation theory
  10. Second-order perturbation theory
  11. Final result
  12. Matching-scheme dependence
  13. Phenomenology
  14. QED
  15. Conclusions
  16. ...and 10 more sections

Figures (4)

  • Figure 1: P-wave hyperfine splitting for $n=2$ bottomonium. Experimental line and error band in red. Theory predictions are carried out with the PV mass $m_{b,\rm PV}=4.836$ GeV quoted from Ayala:2019hkn.
  • Figure 2: P-wave hyperfine splitting for $n=2$$B_c$. Theory predictions are carried out with the PV mass $m_{b,\rm PV}=4.836$ GeV quoted from Ayala:2019hkn and $m_{c,\rm PV} =1.498$ GeV. The latter is obtained using the experimental mass of the $D$ mesons and the value of $\bar{\Lambda}$ determined in Ayala:2019hkn.
  • Figure 3: P-wave hyperfine splitting for $n=2$ charmonium. Experimental line and error band in red. Theory predictions are carried out with the PV mass $m_{c,\rm PV}=1.498$ GeV, obtained using the experimental mass of the $D$ mesons and the value of $\bar{\Lambda}$ determined in Ayala:2019hkn.
  • Figure 4: P-wave hyperfine splitting for $n=3$ bottomonium. Experimental line and error band in red. Theory predictions are carried out with the PV mass $m_{b,\rm PV}=4.836$ GeV quoted from Ayala:2019hkn.