The leading contribution of the light-by-light scattering effects to g-factor
of a bound electron is derived. The corresponding amplitude is expressed in
terms of low-energy Delbruck scattering of a virtual photon. The result reads
Delta g = (7/216) alpha (Z alpha)^5

Alexander I. Milstein

Beier

Costantini

Farnham

Häffner

Karshenboim

Milstein

Mohr

Papatzacos

Savely G. Karshenboim

Shabaev

Yelkhovsky

Yerokhin

Physics Letters B

English

arXiv

Crossref

AbstractThe leading contribution of the light-by-light scattering effects to g-factor of a bound electron is derived. The corresponding amplitude is expressed in terms of low-energy Delbrück scattering of a virtual photon. The result reads Δg=(7/216)α(Zα)5

Karshenboim, Savely G.

Milstein, Alexander I.

Elsevier - Publisher Connector 

Physics Letters B 549 (2002) 321–324
www.elsevier.com/locate/npe
Delbrück scattering and the g-factor of a bound electron
Savely G. Karshenboim a,b, Alexander I. Milstein c
a D.I. Mendeleev Institute for Metrology (VNIIM), St. Petersburg 198005, Russia
b Max-Planck-Institut für Quantenoptik, 85748 Garching, Germany
c Budker Institute of Nuclear Physics, 630090 Novosibirsk, Russia
Received 25 September 2002; received in revised form 25 October 2002; accepted 29 October 2002
Editor: G.F. Giudice
Abstract
The leading contribution of the light-by-light scattering effects to g-factor of a bound electron is derived. The corresponding
amplitude is expressed in terms of low-energy Delbrück scattering of a virtual photon. The result reads g = (7/216)α(Zα)5.
 2002 Elsevier Science B.V.
PACS: 12.20.-m; 31.30.Jv; 32.10.Dk
Keywords: Quantum electrodynamics; g-factor; Delbrück scattering
1. Introduction
Recently a significant progress in measurements of
the g-factor of a bound electron in a hydrogen-like
ion with a spinless nucleus has been achieved [1,2].
At the same time an accurate theory was successfully
developed [3–10]. Comparison of theory and exper-
iment [1] for the hydrogen-like carbon ion 12C5+ al-
lows to determine the most accurate values of the elec-
tron mass [7]
(1)me = 0.000 548 579 909 2(4)u
and the proton-to-electron mass ratio
(2)mp/me = 1836.152 673 3(14).
E-mail addresses: sek@mpq.mpg.de (S.G. Karshenboim),
milstein@inp.nsk.su (A.I. Milstein).
Those are three times more accurate than the recom-
mended CODATA values [11] based on study of pro-
tons and electrons in Penning trap [12].
Theory and experiment equally contribute into un-
certainty of the values in Eqs. (1) and (2). An essential
part of theoretical uncertainty (maybe even the domi-
nant part) is due to the light-by-light scattering effects.
A part of them is related to the Wichmann–Kroll po-
tential (so-called the “electric-loop” term presented in
Fig. 1, where the contribution of a vacuum polarization
by free electrons known analytically for a point-like
nucleus [6] is not included). Its leading contribution
(∝ α(Zα)6) has been found analytically [4]:
(3)g(EL)= 2
(
38
45
− 2π
2
27
)
α(Zα)6
π
.
Here Z is the nuclear charge number, α = e2 is the
fine-structure constant, h¯= c = 1. The other part (so-
called the “magnetic-loop” term) presented in Fig. 2
0370-2693/02  2002 Elsevier Science B.V.
PII: S0370-2693(02)0 29 30 -1
Open access under CC BY license.
Open access under CC BY license.
322 S.G. Karshenboim, A.I. Milstein / Physics Letters B 549 (2002) 321–324
Fig. 1. One of the diagrams for the “electric-loop” contribution
to the g-factor of a bound electron. The internal dashed line
corresponds to D00 component of the photon propagator. The
dashed line with the cross denotes a Coulomb field, the wavy line
with the square denotes the external homogeneous magnetic field.
Fig. 2. One of the diagrams for the “magnetic-loop” contribution to
the g-factor of a bound electron. The internal wavy line corresponds
to Dij component of the photon propagator. Other notations are the
same as in Fig. 1.
has not been known. Even the order of magnitude of
the leading term has not been clarified. Some rough
estimations for it were included into evaluations in
Refs. [3,6,7]. We claim that the magnetic-loop effects
contribute in order α(Zα)5. Therefore, potentially
they could strongly affect the g-factor of a bound
electron. For instance, with
g(ML)
2
= CMLα(Zα)5
one can find a shift of g-factor up to 2CML × 10−9
in the case of the hydrogen-like ion of carbon, and
CML×10−8 in the case of oxygen. That may be bigger
than both experimental and theoretical uncertainty. In
this Letter we study the leading contribution of the
magnetic-loop effects and determine coefficient CML.
Fig. 3. A diagram for Delbrück scattering. The wavy line indicates
here the incoming or outgoing photon.
2. Low-energy Delbrück scattering and the
contribution of “magnetic loop”
First we note that we need to study a vacuum po-
larization loop with the two lines of external virtual
photons in the Coulomb field of a nucleus. One of
these photons is related to an homogenous magnetic
field and the other connects the electron loop and the
atomic electron line. Because of the Furry theorem the
leading effects of the electric field are related to the
diagram with the two Coulomb lines. Then, we apply
the block of the vacuum polarization when both exter-
nal photon lines transfer momenta k1 and k2 signif-
icantly smaller than the electron mass, |k2| ∼ Zαme.
The rigorous analysis shows that the contribution to g-
factor of the region |k2| ∼ me is of order of α(Zα)6.
The kinematics k1, k2 	me is similar to that for low-
energy Delbrück scattering (see Fig. 3).
Delbrück scattering (see [13,14]) is a process in
which the initial photon turns into a virtual electron–
positron pair that scatters in the electric field of an
atom and then transforms into the final photon. The
Feynman diagram corresponding to the amplitude in
the lowest in Zα order is shown in Fig. 3. Let us con-
sider the scattering of a virtual photon with the ini-
tial momentum k1 = (ω,k1) and the final momentum
k2 = (ω,k2), with ω, |k|1,2 	 me. Due to the gauge
invariance, the Delbrück scattering amplitude reads
(4)
T µν = α(Zα)
2
m3e
{
C1 ·
[
gµν(k1 · k2)− kµ2 kν1
]
+C2 ·
[
ω2gµν −ω(nµkν1 + kµ2 nν)
+ (k1 · k2)nµnν
]}
,
S.G. Karshenboim, A.I. Milstein / Physics Letters B 549 (2002) 321–324 323
where nµ = gµ0, C1 and C2 are some constants.
This form of the amplitude follows from the relations
k1µT µν = T νµk2µ = 0, and from the linearity of T µν
with respect to k1 and k2. To calculate the magnetic
loop contribution we need only the amplitude T ij at
ω = 0. We can find it since the amplitude in Eq. (4)
was derived for the real photons. For ω = |k|1 =
|k|2 the amplitude TD = e(1)µ T µνe(2)∗ν (e(1,2)µ are the
polarization vectors) is of the form
(5)
TD = α(Zα)
2
m3e
{−(C1 +C2)ω2(e1 · e∗2)
+C1[e1 × k1][e∗2 × k2]
}
.
The amplitude TD (5) was obtained in Ref. [15] (see
also [16]). Using the results of [15] and Eq. (5), we
obtain
(6)C1 = 716 · 72 and C2 =−
73
32 · 72 .
Let us consider now the amplitude TM of interaction
of the magnetic field with the spin part of the magnetic
moment of the atomic electron. In the zero approxima-
tion it reads
T
(0)
M =−
ie
me
∫
d3k
(2π)3
Ak · [k× s]ρk
(7)= e
me
∫
d3k
(2π)3
Bk · sρk,
where Ak is the vector potential, s is the spin operator,
and ρk is the Fourier transform of the electron den-
sity |ψ(r)|2, ψ(r) is the bound electron wave function.
Note that a sign of TM is opposite to that of Hamil-
tonian. In the case of the homogeneous magnetic field
B, we have
Bk = i[k×Ak] = (2π)3δ(k)B,
and we can replace in Eq. (7) ρk by ρ0 = 1. Using (4)
we can represent the correction T (1)M as follows:
T
(1)
M =−
ieα(Zα)2
m4e
C1
∫
d3k
(2π)3
∫
d3q
(2π)3
4π
q2
× {(k · q)Ak − (Ak · q)k} · [q× s]ρq
=− ieα(Zα)
2
m4e
C1
∫
d3k
(2π)3
Ak · [k× s]
×
∫
d3q
(2π)3
8π
3
ρq
=− ieα(Zα)
2
m4e
C1
(8)×
∫
d3k
(2π)3
Ak · [k× s] · 8π3 |ψ(0)|
2.
Thus, our final result for the relative correction of the
magnetic loop reads
(9)g(ML)
2
= T (1)M /T (0)M =
7πα(Zα)2
6 · 72m3e
|ψ(0)|2.
Substituting |ψ(0)|2 = (Zαm)3/π for 1s state, we
have
(10)g(ML)= 7
216
α(Zα)5.
For the interesting cases of hydrogen-like carbon (Z=
6) and oxygen (Z = 8) the numerical values are 0.4×
10−10 and 1.6 × 10−10, respectively. The obtained
correction has the same order of magnitude as results
in Ref. [3]. However, a quantitative comparison of our
analytic result with numerical calculations there is not
helpful because of lack of accurate numerical data for
Z  10.
3. Conclusions
If one applies Eq. (10) to the g-factor for hydrogen-
like carbon and oxygen, then the result in some sense
is controversial. First, it has lower order of magnitude
than expected, i.e., (α(Zα)5 versus α(Zα)6). On the
other side, the correction (10) is consistent with the
preliminary estimate (see, e.g., [6]) because of its very
small numerical coefficient. Next, it is not clear if the
leading in Zα term gives a dominant contribution.
E.g., in the case of recoil correction for carbon, the
leading term is smaller than the next-to-leading term
because of the small coefficients in it [8,9].
To estimate uncertainty related to the higher-order
terms, we note that the coefficients in the contribution
of the free vacuum polarization [6]
g(VP)= 2 · α
π
·
[
− 8
15 (Zα)
4 + 5π
18
(Zα)5
]
(11)−0.34× α(Zα)5 + 0.56× α(Zα)6
are about unity in contrast to the light-by-light contri-
butions (3) and (10),
g(ML) 0.032× α(Zα)5,
324 S.G. Karshenboim, A.I. Milstein / Physics Letters B 549 (2002) 321–324
Table 1
The bound electron g-factor in low-Z hydrogen-like ions with
spinless nucleus. The uncertainty for the two-loop contribution is
taken from [6]. Ref. (one-loop) is related to the one-loop result for
the self-energy contribution. For lighter atoms it is taken from [17]
based on fitting data of [3], while for heavier isotopes we use the
results of [10]
Ion g Ref. (one-loop)
4He+ 2.002 177 406 7(1) [17]
10Be3+ 2.001 751 574 5(4) [17]
12C5+ 2.001 041 590 1(4) [10]
16O7+ 2.000 047 020 1(8) [10]
(12)g(EL) 0.072× α(Zα)6.
We consider an estimation ±α(Zα)6 for higher order
magnetic-loop effects as a conservative one. It is be-
low 10−10 at Z < 6, leads to g ∼ 1× 10−10 for car-
bon and g ∼ 6 × 10−10 for oxygen. Our results are
collected in the Table 1. The other sources of theoret-
ical uncertainty, as explained in Refs. [6,17], are nu-
merical error of evaluation of the one-loop self-energy
contribution and estimation of unknown higher-order
two-loop terms.
The experimental results for g are available for the
ions of carbon and oxygen with a relative uncertainty
of 2 × 10−9 [1,2] being limited by our knowledge of
the electron mass [12]. However, their ratio is free of
this uncertainty
(13)g(12C5+)/g(16O7+)= 1.000 497 273 1(15),
and is in a fair agreement with the theoretical predic-
tion
(14)g(12C5+)/g(16O7+)= 1.000 497 273 3(3).
Calculation of the next-to-leading term of the magnet-
ic-loop effects is in progress.
Acknowledgements
We are grateful to Peter Mohr, Günther Werth,
Vladimir Ivanov for stimulating discussions. We thank
the School of Physics at the University of New South
Wales for their warm hospitality during the stay when
this work has been done. The work of S.G.K. was
supported in part by RFBR grant 00-02-16718, and the
work of A.I.M. by UNSW.
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Delbrück scattering and the g-factor of a bound electron 

Delbruck scattering and the g-factor of a bound electron

Delbruck scattering and the g-factor of a bound electron

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