1. Nuclear recoil in the Lamb shift of hydrogen-like atoms
Vladimir A. Yerokhin
Peter the Great St. Petersburg Polytechnic University
and
Vladimir Shabaev
St. Petersburg State University
2nd ECT Workshop on the Proton Radius Puzzle, June 21, 2016, Trento 1

2. Outline Nuclear recoil effect for the point nucleus
Nuclear recoil effect for an extended nucleus
Results for hydrogen Lamb shift 2

3. Motivation
A disagreement between the Zα expansion result and the all-order numerical calculations for the higher-order nuclear recoil in the hydrogen Lamb shift.
In units 1/π (Zα)7 m/M:
Zα expansion : -18
All-order:
(difference for 1S ~ 0.65 kHz )
==> Second-largest theoretical uncertainty in the hydrogen Lamb shift. 3

4. Nonrelativistic recoil in atoms: almost trivial4

5. Nonrelativistic recoil, ~(Zα)2
Nonrelativistic recoil operator:
Normal mass shift
Specific mass shift
For one-electron atoms, the nonrelativistic recoil is just the reduced mass.
This is valid to all orders in m/M. 5

6. Leading relativistic recoil, ~ m/M (Zα)4
Relativistic recoil operator:
Shabaev, Sov. J. Nucl. Phys. 1988
Can be derived from the Breit equation.
Valid through order (Zα)4.
For H-like atoms, the leading relativistic recoil correction to energy: 6

7. Higher-order recoil, ~ m/M (Zα)5+
Salpeter, 1955:
Pachucki and Grotch, 1995; Golosov et al. 1995:
Melnikov and Yelkhovsky, 1999; Pachucki and Karshenboim 1999: 7

8. Nuclear recoil, ~ m/M and all orders in (Zα)
“Coulomb part”: exchange by arbitrary number of Coulomb photons between electron and nucleus
“One transverse photon part”: exchange by arbitrary number of Coulomb photons and one transverse photon
“Two transverse photons part”: exchange by arbitrary number of Coulomb photons and two transverse photons 8

9. Nuclear recoil, ~ m/M and all orders in (Zα)
Shabaev, 1988; Pachucki and Grotch 1995;
Yelkhovsky 1994
“Coulomb part”:
“One transverse photon part”:
“Two transverse photons part”: 9

10. All-order (in Zα) calculations (1)
Numerical all-order calculations:
Artemyev, Shabaev and Yerokhin, 1995; Shabaev et al., 1998
Agreement with the first terms of the Zα expansion …
All - order
Zα expansion
(Zα)5 & (Zα)6 10

11. All-order (in Zα) calculations (2)
But in the higher-orders …
The higher-order remainder, all-order calculations:
Artemyev, Shabaev and Yerokhin, 1995
Shabaev et al., 1998
Zα expansion 11

12. Higher-order remainder (1)
Yerokhin and Shabaev, PRL 115, (2015)
More advanced all-order calculations …
All-order, 2S
All-order, 1S
Zα expansion
D72 (fitted) = (1)
D72 (exact) = -11/60 = … 12

13. Higher-order remainder (2)
Yerokhin and Shabaev, PRL 115, (2015)
Fit to the numerical 1s results:
For Z = 1: 13

14. Nuclear size correction to nuclear recoil (1)
The reduced-mass correction to the leading nuclear size correction:
What is next? 14

15. Nuclear size correction to nuclear recoil (2)
Grotch and Yennie, 1967; Borie and Rinker 1982
Recoil correction from the Breit equation with an extended nuclear charge:
Shabaev et al., 1998
The finite nuclear size correction obtained from the Breit equation contains
a large spurious contribution which is exactly cancelled by the Coulomb part ΔEC . 15

16. Nuclear size correction to nuclear recoil (3)
For an extended nucleus, the formula for the Coulomb part is the same as for the point nucleus; V(r) is the nuclear potential.
Extended-nucleus formulas for the one- and two-transverse photon parts are not known. Approximate treatment. 16

17. Results for hydrogen Lamb shift
Results for P(Zα): 1s 2s
Yerokhin & Shabaev 2015
(2)
(2)
(2)
Point
FNS
Shabaev et al. 1998
(3)
(5)
Zα expansion
FNS is the finite nuclear size correction BEYOND the reduced mass. 17

18. H - D isotope shift Experiment, in kHz 670 994 334. 606 (15)
Parthey et al., PRL 104, (2010)
Theory, point nucleus, in kHz
(66)(60)
Jentschura et al. PRA 83, (2011)
Deutron-proton radii difference
(65) fm2
Additional contributions kHz point nucleus
0. 08 kHz finite nuclear size
Increase the deutron-proton radii difference by
fm2 18

19. Conclusion
Nuclear recoil correction ~m/M for hydrogen Lamb shift is calculated to all orders in Zα.
Higher-order remainder is identified. Agreement with Zα expansion results is demonstrated.
A large single-log contribution to order (Zα)7 is found, which changed significantly the higher-order contribution for hydrogen.
The second-largest theoretical uncertainty in the hydrogen Lamb shift is removed. 19