1. The bound-electron g factor in light ions
Vladimir A. Yerokhin
Peter the Great St. Petersburg Polytechnic University
Precision Physics and Fundamental Physical Constants FFK-2017, May 19, Warsaw 1

2. Outline Free-electron g factor. Bound-electron g factor: H-like ions.
Bound-electron g factor: Li-like ions.
Electron mass and fine-structure constant determinations. 2

3. g factor of free electron: Classical picture
What is g factor ?
 It is essentially a proportionality constant that relates the observed magnetic moment μ of a particle to its angular momentum quantum number
Classical electrodynamics: g factor = 1
Relativistic quantum mechanics: g factor = 2 3

4. g factor of free electron: Quantum electrodynamics
Engraved on Julian Schwinger’s tombstone 4

5. g factor of free electron: Quantum electrodynamics
Engraved on Julian Schwinger’s tombstone 5

6. g factor of free electron: Quantum electrodynamics6

7. What do we learn from free-electron g factor?
Best prediction of the (free-field) Quantum Electrodynamics
Best determination of the fine-structure constant
QED is the most successful theory in all of physics 7

8. Bound-electron g factor
Experiments:
Experimental result for 12C5+:
ωL/ωc = (11)(7)
[Sturm et al. Nature 506, 467 (2014)] 8

9. What can we learn from bound-electron g factor?
Tests of bound-state Quantum Electrodynamics Advantages: high experimental accuracy, weak nuclear effects, different Z and charge states
Electron mass
Fine-structure constant
Nuclear magnetic moments, nuclear masses, nuclear radii 9

10. g factor of H-like ions: theory
Effects
Relativistic
Free QED
Bound QED
Nuclear
Nuclear recoil
Nuclear size
Nuclear polarization & deformation 10

11. Leading orders
Dirac g factor:
Free QED, αn (Zα)0: 11

12. Bound QED, Zα expansion
Bound QED, αn (Zα)2 : [H. Grotch 1970, R.N. Faustov 1970, F.E. Close and H. Osborn 1971,…]
Bound QED, 1-loop, α (Zα)4: [K. Pachucki, A. Czarnecki, U.D. Jentschura, VAY, 2005]
Bound QED, 2-loop, α2 (Zα)4: [K. Pachucki, A. Czarnecki, U.D. Jentschura, VAY, 2005]
[Czarnecki
and
Szafron,
PRA, 2016] 12

13. Bound QED, all orders in Zα
Electron Self-energy
Vacuum-polarization
Magnetic-loop
[Karshenboim & Milstein 2002, Lee et al, 2005]
Electric-loop
[Beier et al. PRA 2000] 13

14. So far, an “experimental” determination of two-loop QED effects
only partial results up to now
VAY and Z. Harman PRA 2013
work in progress …
2-loop QED corrections, all orders in Zα:
The main theoretical uncertainty of the g factor !
So far, an “experimental” determination of two-loop QED effects 14

15. Nuclear effects Nuclear recoil Finite nuclear size
to leading order in Zα
[Grotch, PRA 1970, Faustov, PLB 1970]
to all orders in Zα
[Shabaev, PRA 2001; Shabaev and VAY PRL 2002]
[Faustov 1970, Grotch and Hegstrom 1971, Close and Osborn 1971, and others]
Finite nuclear size
[VAY et al. JPB 2003; PRA 2016]
Nuclear polarization and deformation
[Nefiodov et al. PLB 2003, Volotka and Plunien PRL 2014, Zatorski et al. PRL 2012] 15

16. g factor of H-like carbon: present status
Theoretical
estimate
Derived from
experiment
on Si13+
[J. Zatorsky et al. 2017] 16

17. Determination of the electron mass
Bound-electron g factor:
me = (14)(9)(2) (stat)(sys)(theory)
Independent determination (free electron versus C6+):
me = (12) [Farnheim et al. PRL 75, 3598 (1995)] 17

18. g factor of Li-like ions
Parameters:
Methods
Z α expansion,
all orders in 1/Z
1/Z expansion,
all orders in Zα 18

19. g factor of Li-like ions
Parameters:
Methods
Z α expansion,
all orders in 1/Z
1/Z expansion,
all orders in Zα
Unified
theory 19

20. 1/Z expansion, all orders in Zα
g (Li-like)
g (H-like)
Electron-electron interaction
1-photon exchange
2-photon exchange
(>=3)-photon exchange
QED + Electron-electron interaction
QED + 1-photon exchange
QED +(>=2)-photon exchange
[Volotka et al. PRL 2014; Glazov et al. PRA 2010, PRA 2004; Shabaev et al. PRA 2002] 20

21. Zα expansion, all orders in 1/Z
Nonrelativistic Quantum Electrodynamics (NRQED) expansion:
Effective Hamiltonian, describing the interaction of an atom with the external magnetic field to orders α2, α3, and m/M [Hegstrom 73]:
Numerical calculations:
[Yan PRL 2001; JPB 2002]
[VAY, Pachucki, Puchalski et al., arXiv:submit/ ] 21

22. Unified theory Zα expansion 1/Z expansion NRQED, O(α2)
[VAY, Pachucki, Puchalski et al., arXiv:submit/ ]
Electron correlation:
Zα expansion
1/Z
expansion
NRQED, O(α2) 22

23. Binding effects in g factor Li-like silicon, current status23

24. Binding effects in g factor Li-like silicon, current status24

25. g-factor of Li-like calcium. Isotope shift
Theory: only one effect (nuclear recoil) contributes !
Nonrelativistic nuclear recoil vanishes (for s states) !
Test of relativistic theory of the nuclear recoil effect. 25

26. What’s next: Fine-structure constant ?
If we get α from the free-electron, why not from the bound-electron?
Problems: theory is much more complicated
nuclear effects
Advantages: we can vary Z and the charge state 26

27. Fine-structure constant from bound-electron g factor
Possible ways to go:
H-like ions, Z as small as possible (He+)
Li-like + H-like ions, small Z [Yerokhin et al. PRL 116, (2016)]
B-like + H-like ions, large Z [Shabaev et al. PRL 96, (2006)] 27

28. Weighted difference of H-like and Li-like ions, Low Z.
[Yerokhin et al. PRL 116, (2016)]
Li-like ion
Difference
Nuclear effects are suppressed by three orders of magnitude.
They do not create problems for α determination.
Nuclear effects are not a problem ! 28

29. Outlook: Determination of nuclear magnetic moments
g factor of an ion with a spin-nonzero nucleus (I – nuclear spin, J – electron angular momentum, F – electron + nucleus angular momentum):
All corrections can be parameterized in terms of the shielding constant σ:
Shielding in H-like ions is calculated up to a very high accuracy.
[Yerokhin et al. PRL 2011] 29

30. Conclusion
Bound-electron g factor: high-precision measurements and accurate calculations.
Tests of bound-state QED
Determination of the electron mass
In future: access to the fine-structure constant, nuclear magnetic moments, nuclear masses, nuclear charge radii 30

31. THANK YOU FOR YOUR ATTENTION! Conclusion
Bound-electron g factor: high-precision measurements and accurate calculations.
Tests of bound-state QED
Determination of the electron mass
In future: access to the fine-structure constant, nuclear magnetic moments, nuclear masses, nuclear charge radii
THANK YOU
FOR YOUR ATTENTION! 31

32. Additional
Bound-electron g factor: high-precision measurements and accurate calculations.
Tests of bound-state QED
Determination of the electron mass
Access to the fine-structure constant, nuclear magnetic moments, nuclear masses, nuclear charge radii 32