Precise theory of levels of hydrogen and deuterium:

The two-photon radiative corrections

Ulrich D. Jentschura1, 2 and Peter J. Mohr1

1National Institute of Standards and Technology, Gaithersburg, MD 20899, USA

2Max–Planck–Institut fur Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany  ̈

(Dated: April 20, 2005)

The two-photon radiative effects are significant quantum electrodynamic corrections to the energy levels of

hydrogen and deuterium atoms. Calculations of higher-order contributions to this correction are reviewed, and

the results needed to evaluate energy levels for states with n ≤ 200 are given. The results of such an evaluation

are available on the NIST Physics Laboratory Web site at physics.nist.gov/hdel.

PACS numbers: 12.20.Ds, 31.30.Jv, 06.20.Jr, 31.15.-p

I. INTRODUCTION

Energy levels in hydrogen and deuterium are determined

primarily by the eigenvalues of the Dirac equation, but to obtain accurate values for the levels, it is necessary to include

many additional corrections including those due to quantum

electrodynamic (QED) effects. In hydrogen and deuterium,

the two-photon radiative correction is a significant QED effect.

II. TWO-PHOTON CORRECTIONS

Corrections from two virtual photons, of order α

2

, have

been calculated as a power series in Zα:

E

(4) =

α

π

2 (Zα)

4

n3 mec

2F(4)(Zα) , (1)

where

F

(4)(Zα) = B40 + B50 (Zα) + B63 (Zα)2

ln3(Zα)

−2

+B62 (Zα)

2

ln2(Zα)

−2

+B61 (Zα)

2

ln(Zα)

−2 + B60 (Zα)2

+ · · · . (2)

The leading terms B40, B50, B63, and B62 are known for

all states and are discussed in Appendix A of Ref. [1]. References to calculations of these terms are also given there. In

this note, we are concerned with the information available for

the coefficients B61 and B60

III. THE COEFFICIENT B61

The single-logarithm coefficient B61 for S states has been

given as [2]

B61 =

39 751

10 800

+

4N(n)

3

+

55π

2

27

−

616 ln 2

135

+

3π

2

ln 2

4

+

40 ln22

9

−

9 ζ(3)

8

+



304

135

−

32 ln 2

9



×



3

4

+ γ + ψ(n) − ln n −

1

n

+

1

4n2



, (3)

TABLE I: Values of N for S states in this work

n N

1 17.855 672(1)

2 12.032 209(1)

3 10.449 810(1)

4 9.722 413(1)

5 9.304 114(1)

6 9.031 832(1)

7 8.840 123(1)

8 8.697 639(1)

9 8.2(8)

10 8.2(8)

11 8.1(8)

12 8.1(8)

13 8.1(8)

14 8.0(8)

15 8.0(8)

where N(n) is a term that was numerically evaluated for the

1S state by Pachucki [2]. Jentschura [3] has evaluated N(n)

for excited S states with n = 2 to n = 8, has made an improved evaluation for n = 1, and has given an approximate fit

to the calculated results in order to extend them to higher n.

The fitted function for S states is

N(n) = 7.78 +

3.13

n

+

6.93

n2

+ · · · (4)

There are no complete results yet for P, D, or higher-l states

for B61, although the term N(n) has been calculated for P

states in Ref. [3].

Values of the function N(n) for some S states are given in

Table I. The values for n ≥ 9 are based on Eq. (4), with an

assumed uncertainty of type un of 10 % of the quoted value.

Based on the relative magnitude of A61 for the S, P, and D

states, we take as uncertainties un(B61) = 5.0 for P states

and un(B61) = 0.5 for D and higher-l states.

Recent work indicates that there may be an additional contribution to B61 and/or B60 [4, 5]. The effect of such a contribution would be to change the S-state energy levels by an

amount that is likely to be less than half the uncertainty of

the nuclear size correction due to uncertainty in the rms radius of the nucleus. However, a change of this size in either

coefficient that is independent of n would have a relatively
2

TABLE II: Values of bL and B60 used in the 2002 adjustment

n bL B60

1 −81.4(3) −61.6(9.2)

2 −66.6(3) −53.2(8.0)

3 −61.7(5.0) −50.1(9.0)

4 −59.2(5.0) −48.4(8.8)

5 −57.7(5.0) −47.4(8.7)

6 −56.7(5.0) −46.7(8.6)

7 −56.0(5.0) −46.2(8.5)

8 −55.5(5.0) −45.8(8.5)

9 −55.1(5.0) −46.0(8.6)

10 −54.8(5.0) −45.7(8.5)

11 −54.5(5.0) −45.5(8.5)

12 −54.3(5.0) −45.3(8.5)

13 −54.1(5.0) −45.1(8.5)

14 −53.9(5.0) −45.0(8.4)

15 −53.8(5.0) −44.9(8.4)

minor effect on the calculated energy levels, based on a leastsquares adjustment. The reason is that such a new coefficient,

when included in the least-squares adjustment of the constants

would lead to a new adjusted value of the nuclear charge radius, since both corrections to the energy level are proportional to 1/n3

, and the sum of the corrections is determined

by the experimental data. Thus, the change in the coefficient

and the change in the nuclear size correction would essentially

cancel in the net contribution to the energy levels.

IV. THE COEFFICIENT B60

The two-loop Bethe logarithm bL, which is expected to be

the dominant part of the no-log term B60, has been calculated

for the 1S and 2S states by Pachucki and Jentschura [6] who

obtained

bL = −81.4(3) 1S state (5)

bL = −66.6(3) 2S state . (6)

An additional contribution for S states,

bM =

10

9

N , (7)

as derived by Pachucki [2], where N is given in Table I as a

function of the state n.

These contributions can be combined to obtain an estimate

for the coefficient B60 for S states:

B60 = bL +

10

9

N + · · · , (8)

where the dots represent uncalculated contributions to B60

which are at the relative level of 15 % [6].

In order to obtain an approximate value for B60 for S states

with n ≥ 3, we employ a simple extrapolation formula,

bL = a +

b

n

, (9)

with a and b fitted to the 1S and 2S values of bL, and we

include a component of uncertainty u0(bL) = 5.0. The results for bL, along with the total estimated values of B60 for S

states, is given in Table II. For P states, there is a calculation

of fine-structure differences [7], but because of the uncertainty

in B61 for P states, we do not include this result. We assume

that for both the P and D states, the uncertainty attributed to

B61 is sufficiently large to account for the uncertainty in B60

and higher-order terms as well.

[1] P. J. Mohr and B. N. Taylor, Rev. Mod. Phys. (2005), in press;

physics.nist.gov/constants.

[2] K. Pachucki, Phys. Rev. A 63, 042503 (2001).

[3] U. D. Jentschura, J. Phys. A 36, L229 (2003).

[4] K. Pachucki, private communication (2004).

[5] V. A. Yerokhin, P. Indelicato, and V. M. Shabaev, hepph/0409048 (2004).

[6] K. Pachucki and U. D. Jentschura, Phys. Rev. Lett. 91, 113005

(2003).

[7] U. D. Jentschura and K. Pachucki, J. Phys. A 35, 1927 (2002).